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Properties

Label 4-3264e2-1.1-c1e2-0-19
Degree 44
Conductor 1065369610653696
Sign 11
Analytic cond. 679.288679.288
Root an. cond. 5.105215.10521
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 4·13-s − 2·17-s − 8·19-s + 6·25-s − 16·43-s + 10·49-s − 4·53-s − 8·59-s − 24·67-s + 81-s − 24·83-s − 12·89-s − 12·101-s + 4·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 14·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.10·13-s − 0.485·17-s − 1.83·19-s + 6/5·25-s − 2.43·43-s + 10/7·49-s − 0.549·53-s − 1.04·59-s − 2.93·67-s + 1/9·81-s − 2.63·83-s − 1.27·89-s − 1.19·101-s + 0.369·117-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + ⋯

Functional equation

Λ(s)=(10653696s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10653696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10653696s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10653696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1065369610653696    =    212⋅32⋅1722^{12} \cdot 3^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 679.288679.288
Root analytic conductor: 5.105215.10521
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 10653696, ( :1/2,1/2), 1)(4,\ 10653696,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
3C2C_2 1+T2 1 + T^{2}
17C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
29C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
31C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
43C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
67C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
71C22C_2^2 1−138T2+p2T4 1 - 138 T^{2} + p^{2} T^{4}
73C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
79C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
83C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C22C_2^2 1−178T2+p2T4 1 - 178 T^{2} + p^{2} T^{4}
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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

−8.428602581108460320200220546494, −8.308214659363619700251168385294, −7.62681910073560329533321596098, −7.34927671293210359354178302810, −6.91196014220385073134643908580, −6.69344832447220363833996743583, −6.16862100960228138437264922416, −5.93011284460378139820141700956, −5.36373660397584487843703308909, −4.95037755790314960174178662396, −4.50696747772029112664654524817, −4.39562714406982276701382240793, −3.75195544275812733848703153186, −3.20349456927637199665081096295, −2.63384933237430905026090025816, −2.53177668592740285158076870180, −1.71970836998179909189495326566, −1.33236530530530222750183972939, 0, 0, 1.33236530530530222750183972939, 1.71970836998179909189495326566, 2.53177668592740285158076870180, 2.63384933237430905026090025816, 3.20349456927637199665081096295, 3.75195544275812733848703153186, 4.39562714406982276701382240793, 4.50696747772029112664654524817, 4.95037755790314960174178662396, 5.36373660397584487843703308909, 5.93011284460378139820141700956, 6.16862100960228138437264922416, 6.69344832447220363833996743583, 6.91196014220385073134643908580, 7.34927671293210359354178302810, 7.62681910073560329533321596098, 8.308214659363619700251168385294, 8.428602581108460320200220546494

Graph of the ZZ-function along the critical line