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Properties

Label 4-1440e2-1.1-c3e2-0-11
Degree 44
Conductor 20736002073600
Sign 11
Analytic cond. 7218.667218.66
Root an. cond. 9.217529.21752
Motivic weight 33
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  + 10·5-s + 104·13-s + 132·17-s + 75·25-s − 92·29-s − 48·37-s + 144·41-s − 346·49-s − 124·53-s − 380·61-s + 1.04e3·65-s + 2.15e3·73-s + 1.32e3·85-s − 2.23e3·89-s + 1.66e3·97-s − 540·101-s + 3.49e3·109-s − 1.76e3·113-s + 398·121-s + 500·125-s + 127-s + 131-s + 137-s + 139-s − 920·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.894·5-s + 2.21·13-s + 1.88·17-s + 3/5·25-s − 0.589·29-s − 0.213·37-s + 0.548·41-s − 1.00·49-s − 0.321·53-s − 0.797·61-s + 1.98·65-s + 3.45·73-s + 1.68·85-s − 2.65·89-s + 1.74·97-s − 0.532·101-s + 3.06·109-s − 1.46·113-s + 0.299·121-s + 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.526·145-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

Λ(s)=(2073600s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2073600s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20736002073600    =    210⋅34⋅522^{10} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 7218.667218.66
Root analytic conductor: 9.217529.21752
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2073600, ( :3/2,3/2), 1)(4,\ 2073600,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.9685058315.968505831
L(12)L(\frac12) ≈\approx 5.9685058315.968505831
L(52)L(\frac{5}{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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
5C1C_1 (1−pT)2 ( 1 - p T )^{2}
good7C22C_2^2 1+346T2+p6T4 1 + 346 T^{2} + p^{6} T^{4}
11C22C_2^2 1−398T2+p6T4 1 - 398 T^{2} + p^{6} T^{4}
13C2C_2 (1−4pT+p3T2)2 ( 1 - 4 p T + p^{3} T^{2} )^{2}
17C2C_2 (1−66T+p3T2)2 ( 1 - 66 T + p^{3} T^{2} )^{2}
19C2C_2 (1+p3T2)2 ( 1 + p^{3} T^{2} )^{2}
23C22C_2^2 1+22974T2+p6T4 1 + 22974 T^{2} + p^{6} T^{4}
29C2C_2 (1+46T+p3T2)2 ( 1 + 46 T + p^{3} T^{2} )^{2}
31C22C_2^2 1−7058T2+p6T4 1 - 7058 T^{2} + p^{6} T^{4}
37C2C_2 (1+24T+p3T2)2 ( 1 + 24 T + p^{3} T^{2} )^{2}
41C2C_2 (1−72T+p3T2)2 ( 1 - 72 T + p^{3} T^{2} )^{2}
43C2C_2 (1+p3T2)2 ( 1 + p^{3} T^{2} )^{2}
47C22C_2^2 1+97486T2+p6T4 1 + 97486 T^{2} + p^{6} T^{4}
53C2C_2 (1+62T+p3T2)2 ( 1 + 62 T + p^{3} T^{2} )^{2}
59C22C_2^2 1+353298T2+p6T4 1 + 353298 T^{2} + p^{6} T^{4}
61C2C_2 (1+190T+p3T2)2 ( 1 + 190 T + p^{3} T^{2} )^{2}
67C22C_2^2 1+1766T2+p6T4 1 + 1766 T^{2} + p^{6} T^{4}
71C22C_2^2 1+322782T2+p6T4 1 + 322782 T^{2} + p^{6} T^{4}
73C2C_2 (1−1078T+p3T2)2 ( 1 - 1078 T + p^{3} T^{2} )^{2}
79C22C_2^2 1+266638T2+p6T4 1 + 266638 T^{2} + p^{6} T^{4}
83C22C_2^2 1+543814T2+p6T4 1 + 543814 T^{2} + p^{6} T^{4}
89C2C_2 (1+1116T+p3T2)2 ( 1 + 1116 T + p^{3} T^{2} )^{2}
97C2C_2 (1−834T+p3T2)2 ( 1 - 834 T + p^{3} T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.292408489467636932695143633122, −9.094074067100686678810221388771, −8.434165868490913458985467548450, −8.333862725516490303294986211629, −7.64658497517380748147937196771, −7.59383158208400771342068979637, −6.65282927931518761678220750989, −6.55991780345456703010106489804, −5.98380194938409960625172545185, −5.76319296250611737459743097658, −5.28014519735133870496154064321, −4.95423108890951503159742496235, −4.08652211306604070328189083540, −3.84065923310939226015173632909, −3.11029629006701676661297624103, −3.08980966966120736478202463856, −2.03099962148204467376287099386, −1.64130278522732706370938916534, −1.08737333887223422561925363929, −0.61870669156209468842811600692, 0.61870669156209468842811600692, 1.08737333887223422561925363929, 1.64130278522732706370938916534, 2.03099962148204467376287099386, 3.08980966966120736478202463856, 3.11029629006701676661297624103, 3.84065923310939226015173632909, 4.08652211306604070328189083540, 4.95423108890951503159742496235, 5.28014519735133870496154064321, 5.76319296250611737459743097658, 5.98380194938409960625172545185, 6.55991780345456703010106489804, 6.65282927931518761678220750989, 7.59383158208400771342068979637, 7.64658497517380748147937196771, 8.333862725516490303294986211629, 8.434165868490913458985467548450, 9.094074067100686678810221388771, 9.292408489467636932695143633122

Graph of the ZZ-function along the critical line