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Properties

Label 4-2e16-1.1-c7e2-0-2
Degree 44
Conductor 6553665536
Sign 11
Analytic cond. 6395.296395.29
Root an. cond. 8.942628.94262
Motivic weight 77
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  + 912·7-s − 2.68e3·9-s − 2.23e4·17-s − 1.63e5·23-s + 1.49e5·25-s − 8.09e4·31-s − 2.82e5·41-s − 1.36e6·47-s − 1.02e6·49-s − 2.44e6·63-s + 5.09e6·71-s + 3.36e6·73-s + 8.07e6·79-s + 2.41e6·81-s + 1.29e7·89-s − 1.21e7·97-s − 8.20e6·103-s + 1.86e7·113-s − 2.03e7·119-s + 3.26e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 5.98e7·153-s + ⋯
L(s)  = 1  + 1.00·7-s − 1.22·9-s − 1.10·17-s − 2.80·23-s + 1.91·25-s − 0.488·31-s − 0.640·41-s − 1.91·47-s − 1.24·49-s − 1.23·63-s + 1.68·71-s + 1.01·73-s + 1.84·79-s + 0.503·81-s + 1.94·89-s − 1.34·97-s − 0.739·103-s + 1.21·113-s − 1.10·119-s + 1.67·121-s + 1.35·153-s − 2.81·161-s + ⋯

Functional equation

Λ(s)=(65536s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(65536s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6553665536    =    2162^{16}
Sign: 11
Analytic conductor: 6395.296395.29
Root analytic conductor: 8.942628.94262
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 65536, ( :7/2,7/2), 1)(4,\ 65536,\ (\ :7/2, 7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 0.44900139660.4490013966
L(12)L(\frac12) ≈\approx 0.44900139660.4490013966
L(92)L(\frac{9}{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
good3C22C_2^2 1+298p2T2+p14T4 1 + 298 p^{2} T^{2} + p^{14} T^{4}
5C22C_2^2 1−149526T2+p14T4 1 - 149526 T^{2} + p^{14} T^{4}
7C2C_2 (1−456T+p7T2)2 ( 1 - 456 T + p^{7} T^{2} )^{2}
11C22C_2^2 1−32603766T2+p14T4 1 - 32603766 T^{2} + p^{14} T^{4}
13C22C_2^2 1−9331750T2+p14T4 1 - 9331750 T^{2} + p^{14} T^{4}
17C2C_2 (1+11150T+p7T2)2 ( 1 + 11150 T + p^{7} T^{2} )^{2}
19C22C_2^2 1−1770736102T2+p14T4 1 - 1770736102 T^{2} + p^{14} T^{4}
23C2C_2 (1+81704T+p7T2)2 ( 1 + 81704 T + p^{7} T^{2} )^{2}
29C22C_2^2 1−24540111814T2+p14T4 1 - 24540111814 T^{2} + p^{14} T^{4}
31C2C_2 (1+40480T+p7T2)2 ( 1 + 40480 T + p^{7} T^{2} )^{2}
37C22C_2^2 1−13932162902T2+p14T4 1 - 13932162902 T^{2} + p^{14} T^{4}
41C2C_2 (1+141402T+p7T2)2 ( 1 + 141402 T + p^{7} T^{2} )^{2}
43C22C_2^2 1−66946399030T2+p14T4 1 - 66946399030 T^{2} + p^{14} T^{4}
47C2C_2 (1+682032T+p7T2)2 ( 1 + 682032 T + p^{7} T^{2} )^{2}
53C22C_2^2 1+937974602250T2+p14T4 1 + 937974602250 T^{2} + p^{14} T^{4}
59C22C_2^2 1−4044092872854T2+p14T4 1 - 4044092872854 T^{2} + p^{14} T^{4}
61C22C_2^2 1−2722187643142T2+p14T4 1 - 2722187643142 T^{2} + p^{14} T^{4}
67C22C_2^2 1−3325050217222T2+p14T4 1 - 3325050217222 T^{2} + p^{14} T^{4}
71C2C_2 (1−2548232T+p7T2)2 ( 1 - 2548232 T + p^{7} T^{2} )^{2}
73C2C_2 (1−1680326T+p7T2)2 ( 1 - 1680326 T + p^{7} T^{2} )^{2}
79C2C_2 (1−4038064T+p7T2)2 ( 1 - 4038064 T + p^{7} T^{2} )^{2}
83C22C_2^2 1−25265648115558T2+p14T4 1 - 25265648115558 T^{2} + p^{14} T^{4}
89C2C_2 (1−6473046T+p7T2)2 ( 1 - 6473046 T + p^{7} T^{2} )^{2}
97C2C_2 (1+6065758T+p7T2)2 ( 1 + 6065758 T + p^{7} 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

−11.14925570086343818736714994976, −10.72532125753540279067780307607, −10.04218243679370980611432845593, −9.577656471168676227115989819965, −9.004117622771594297649082735016, −8.453648364449749936029808142988, −8.105087292105323844963876075342, −7.921895020352833980086120407616, −6.98445456768812272635542563173, −6.40310702394613329366114798368, −6.17195982811732043725960572218, −5.24091705598414179955281739247, −5.01206895375848133957518027066, −4.40262392439682416425100183962, −3.64416350855205688470013100548, −3.12577401834294823773825107298, −2.12233151472190567563808185408, −2.05363110490017659800337975830, −1.09218341792945546554962518828, −0.16054144682128735812978866171, 0.16054144682128735812978866171, 1.09218341792945546554962518828, 2.05363110490017659800337975830, 2.12233151472190567563808185408, 3.12577401834294823773825107298, 3.64416350855205688470013100548, 4.40262392439682416425100183962, 5.01206895375848133957518027066, 5.24091705598414179955281739247, 6.17195982811732043725960572218, 6.40310702394613329366114798368, 6.98445456768812272635542563173, 7.921895020352833980086120407616, 8.105087292105323844963876075342, 8.453648364449749936029808142988, 9.004117622771594297649082735016, 9.577656471168676227115989819965, 10.04218243679370980611432845593, 10.72532125753540279067780307607, 11.14925570086343818736714994976

Graph of the ZZ-function along the critical line