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Properties

Label 12-325e6-1.1-c5e6-0-0
Degree 1212
Conductor 1.178×10151.178\times 10^{15}
Sign 11
Analytic cond. 2.00568×10102.00568\times 10^{10}
Root an. cond. 7.219747.21974
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 106·4-s + 586·9-s − 1.23e3·11-s + 4.84e3·16-s + 6.72e3·19-s − 4.61e3·29-s − 1.27e4·31-s + 6.21e4·36-s + 1.65e4·41-s − 1.30e5·44-s + 3.42e4·49-s − 9.50e3·59-s − 1.73e5·61-s + 8.52e4·64-s − 2.46e5·71-s + 7.12e5·76-s + 2.21e5·79-s + 7.82e4·81-s + 2.24e5·89-s − 7.21e5·99-s + 6.09e4·101-s − 6.08e5·109-s − 4.88e5·116-s − 1.51e4·121-s − 1.35e6·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 3.31·4-s + 2.41·9-s − 3.06·11-s + 4.72·16-s + 4.27·19-s − 1.01·29-s − 2.38·31-s + 7.98·36-s + 1.53·41-s − 10.1·44-s + 2.03·49-s − 0.355·59-s − 5.97·61-s + 2.60·64-s − 5.80·71-s + 14.1·76-s + 3.99·79-s + 1.32·81-s + 3.00·89-s − 7.40·99-s + 0.594·101-s − 4.90·109-s − 3.37·116-s − 0.0943·121-s − 7.89·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯

Functional equation

Λ(s)=((512⋅136)s/2 ΓC(s)6 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((512⋅136)s/2 ΓC(s+5/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 512⋅1365^{12} \cdot 13^{6}
Sign: 11
Analytic conductor: 2.00568×10102.00568\times 10^{10}
Root analytic conductor: 7.219747.21974
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 512⋅136, ( :[5/2]6), 1)(12,\ 5^{12} \cdot 13^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )

Particular Values

L(3)L(3) ≈\approx 7.1708443677.170844367
L(12)L(\frac12) ≈\approx 7.1708443677.170844367
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
13 (1+p4T2)3 ( 1 + p^{4} T^{2} )^{3}
good2 1−53pT2+6393T4−62375p2T6+6393p10T8−53p21T10+p30T12 1 - 53 p T^{2} + 6393 T^{4} - 62375 p^{2} T^{6} + 6393 p^{10} T^{8} - 53 p^{21} T^{10} + p^{30} T^{12}
3 1−586T2+265111T4−7921772p2T6+265111p10T8−586p20T10+p30T12 1 - 586 T^{2} + 265111 T^{4} - 7921772 p^{2} T^{6} + 265111 p^{10} T^{8} - 586 p^{20} T^{10} + p^{30} T^{12}
7 1−34250T2+1157349791T4−19859968898444T6+1157349791p10T8−34250p20T10+p30T12 1 - 34250 T^{2} + 1157349791 T^{4} - 19859968898444 T^{6} + 1157349791 p^{10} T^{8} - 34250 p^{20} T^{10} + p^{30} T^{12}
11 (1+56pT+576781T2+198710800T3+576781p5T4+56p11T5+p15T6)2 ( 1 + 56 p T + 576781 T^{2} + 198710800 T^{3} + 576781 p^{5} T^{4} + 56 p^{11} T^{5} + p^{15} T^{6} )^{2}
17 1−5346778T2+14107385735535T4−24198875660122621868T6+14107385735535p10T8−5346778p20T10+p30T12 1 - 5346778 T^{2} + 14107385735535 T^{4} - 24198875660122621868 T^{6} + 14107385735535 p^{10} T^{8} - 5346778 p^{20} T^{10} + p^{30} T^{12}
19 (1−3360T+9980613T2−16920687808T3+9980613p5T4−3360p10T5+p15T6)2 ( 1 - 3360 T + 9980613 T^{2} - 16920687808 T^{3} + 9980613 p^{5} T^{4} - 3360 p^{10} T^{5} + p^{15} T^{6} )^{2}
23 1−17947666T2+177782686483887T4− 1 - 17947666 T^{2} + 177782686483887 T^{4} - 12 ⁣⋯ ⁣9212\!\cdots\!92T6+177782686483887p10T8−17947666p20T10+p30T12 T^{6} + 177782686483887 p^{10} T^{8} - 17947666 p^{20} T^{10} + p^{30} T^{12}
29 (1+2306T+63301891T2+95048382380T3+63301891p5T4+2306p10T5+p15T6)2 ( 1 + 2306 T + 63301891 T^{2} + 95048382380 T^{3} + 63301891 p^{5} T^{4} + 2306 p^{10} T^{5} + p^{15} T^{6} )^{2}
31 (1+6380T+63602873T2+216555662264T3+63602873p5T4+6380p10T5+p15T6)2 ( 1 + 6380 T + 63602873 T^{2} + 216555662264 T^{3} + 63602873 p^{5} T^{4} + 6380 p^{10} T^{5} + p^{15} T^{6} )^{2}
37 1−196213586T2+17914233453773591T4− 1 - 196213586 T^{2} + 17914233453773591 T^{4} - 12 ⁣⋯ ⁣8812\!\cdots\!88T6+17914233453773591p10T8−196213586p20T10+p30T12 T^{6} + 17914233453773591 p^{10} T^{8} - 196213586 p^{20} T^{10} + p^{30} T^{12}
41 (1−8286T+210623751T2−2260028786820T3+210623751p5T4−8286p10T5+p15T6)2 ( 1 - 8286 T + 210623751 T^{2} - 2260028786820 T^{3} + 210623751 p^{5} T^{4} - 8286 p^{10} T^{5} + p^{15} T^{6} )^{2}
43 1−639846330T2+198848638114826151T4− 1 - 639846330 T^{2} + 198848638114826151 T^{4} - 36 ⁣⋯ ⁣5636\!\cdots\!56T6+198848638114826151p10T8−639846330p20T10+p30T12 T^{6} + 198848638114826151 p^{10} T^{8} - 639846330 p^{20} T^{10} + p^{30} T^{12}
47 1−1108811578T2+563976871326771567T4− 1 - 1108811578 T^{2} + 563976871326771567 T^{4} - 16 ⁣⋯ ⁣8016\!\cdots\!80T6+563976871326771567p10T8−1108811578p20T10+p30T12 T^{6} + 563976871326771567 p^{10} T^{8} - 1108811578 p^{20} T^{10} + p^{30} T^{12}
53 1−424381554T2+113706106857181431T4− 1 - 424381554 T^{2} + 113706106857181431 T^{4} - 92 ⁣⋯ ⁣5292\!\cdots\!52T6+113706106857181431p10T8−424381554p20T10+p30T12 T^{6} + 113706106857181431 p^{10} T^{8} - 424381554 p^{20} T^{10} + p^{30} T^{12}
59 (1+4752T+763206237T2+23933901228768T3+763206237p5T4+4752p10T5+p15T6)2 ( 1 + 4752 T + 763206237 T^{2} + 23933901228768 T^{3} + 763206237 p^{5} T^{4} + 4752 p^{10} T^{5} + p^{15} T^{6} )^{2}
61 (1+86894T+3218447267T2+88402029353204T3+3218447267p5T4+86894p10T5+p15T6)2 ( 1 + 86894 T + 3218447267 T^{2} + 88402029353204 T^{3} + 3218447267 p^{5} T^{4} + 86894 p^{10} T^{5} + p^{15} T^{6} )^{2}
67 1+931485438T2+2776490132958986535T4+ 1 + 931485438 T^{2} + 2776490132958986535 T^{4} + 14 ⁣⋯ ⁣3614\!\cdots\!36T6+2776490132958986535p10T8+931485438p20T10+p30T12 T^{6} + 2776490132958986535 p^{10} T^{8} + 931485438 p^{20} T^{10} + p^{30} T^{12}
71 (1+123252T+7672133745T2+347878540025160T3+7672133745p5T4+123252p10T5+p15T6)2 ( 1 + 123252 T + 7672133745 T^{2} + 347878540025160 T^{3} + 7672133745 p^{5} T^{4} + 123252 p^{10} T^{5} + p^{15} T^{6} )^{2}
73 1−6548382410T2+25607390713292226047T4− 1 - 6548382410 T^{2} + 25607390713292226047 T^{4} - 63 ⁣⋯ ⁣8063\!\cdots\!80T6+25607390713292226047p10T8−6548382410p20T10+p30T12 T^{6} + 25607390713292226047 p^{10} T^{8} - 6548382410 p^{20} T^{10} + p^{30} T^{12}
79 (1−110696T+10847564381T2−593275027409072T3+10847564381p5T4−110696p10T5+p15T6)2 ( 1 - 110696 T + 10847564381 T^{2} - 593275027409072 T^{3} + 10847564381 p^{5} T^{4} - 110696 p^{10} T^{5} + p^{15} T^{6} )^{2}
83 1−16170291778T2+ 1 - 16170291778 T^{2} + 12 ⁣⋯ ⁣1512\!\cdots\!15T4− T^{4} - 61 ⁣⋯ ⁣6461\!\cdots\!64T6+ T^{6} + 12 ⁣⋯ ⁣1512\!\cdots\!15p10T8−16170291778p20T10+p30T12 p^{10} T^{8} - 16170291778 p^{20} T^{10} + p^{30} T^{12}
89 (1−112210T+13708842775T2−819892069403740T3+13708842775p5T4−112210p10T5+p15T6)2 ( 1 - 112210 T + 13708842775 T^{2} - 819892069403740 T^{3} + 13708842775 p^{5} T^{4} - 112210 p^{10} T^{5} + p^{15} T^{6} )^{2}
97 1−29962994618T2+ 1 - 29962994618 T^{2} + 40 ⁣⋯ ⁣7940\!\cdots\!79T4− T^{4} - 37 ⁣⋯ ⁣2837\!\cdots\!28T6+ T^{6} + 40 ⁣⋯ ⁣7940\!\cdots\!79p10T8−29962994618p20T10+p30T12 p^{10} T^{8} - 29962994618 p^{20} T^{10} + p^{30} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.37853881216288424252125835570, −5.36248492467707382710155723171, −5.23113297508819732677710988640, −4.98317167694896253399841079879, −4.65009716430432302613886380304, −4.42326549095289092694903424622, −4.41416057933997276156081313444, −3.89358309487180517708316339660, −3.78214763469832472882748376347, −3.76098703836354212517163431541, −3.16697511899316000262628465123, −2.99868734844195749244487282668, −2.97031795652319445311268804565, −2.78728132722754434436462039183, −2.63213673373330264577622190552, −2.59439118495206476384264718433, −1.97305676170305622553623191302, −1.83107112656120550957677172220, −1.74781311710972457969136815359, −1.49729564351410859940226877168, −1.41622499796912474082293003719, −1.17904378966520690069092343914, −0.69225113690646767743213188588, −0.50602691590769484350880232678, −0.13160402507789351749944687784, 0.13160402507789351749944687784, 0.50602691590769484350880232678, 0.69225113690646767743213188588, 1.17904378966520690069092343914, 1.41622499796912474082293003719, 1.49729564351410859940226877168, 1.74781311710972457969136815359, 1.83107112656120550957677172220, 1.97305676170305622553623191302, 2.59439118495206476384264718433, 2.63213673373330264577622190552, 2.78728132722754434436462039183, 2.97031795652319445311268804565, 2.99868734844195749244487282668, 3.16697511899316000262628465123, 3.76098703836354212517163431541, 3.78214763469832472882748376347, 3.89358309487180517708316339660, 4.41416057933997276156081313444, 4.42326549095289092694903424622, 4.65009716430432302613886380304, 4.98317167694896253399841079879, 5.23113297508819732677710988640, 5.36248492467707382710155723171, 5.37853881216288424252125835570

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.