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Properties

Label 2-70-1.1-c7-0-9
Degree 22
Conductor 7070
Sign −1-1
Analytic cond. 21.866921.8669
Root an. cond. 4.676214.67621
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 93·3-s + 64·4-s + 125·5-s − 744·6-s + 343·7-s + 512·8-s + 6.46e3·9-s + 1.00e3·10-s − 2.16e3·11-s − 5.95e3·12-s − 1.66e3·13-s + 2.74e3·14-s − 1.16e4·15-s + 4.09e3·16-s − 3.57e4·17-s + 5.16e4·18-s + 2.02e4·19-s + 8.00e3·20-s − 3.18e4·21-s − 1.73e4·22-s − 4.21e4·23-s − 4.76e4·24-s + 1.56e4·25-s − 1.32e4·26-s − 3.97e5·27-s + 2.19e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.98·3-s + 1/2·4-s + 0.447·5-s − 1.40·6-s + 0.377·7-s + 0.353·8-s + 2.95·9-s + 0.316·10-s − 0.490·11-s − 0.994·12-s − 0.209·13-s + 0.267·14-s − 0.889·15-s + 1/4·16-s − 1.76·17-s + 2.08·18-s + 0.676·19-s + 0.223·20-s − 0.751·21-s − 0.347·22-s − 0.722·23-s − 0.703·24-s + 1/5·25-s − 0.148·26-s − 3.88·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 21.866921.8669
Root analytic conductor: 4.676214.67621
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 70, ( :7/2), −1)(2,\ 70,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
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}
ppFp(T)F_p(T)
bad2 1−p3T 1 - p^{3} T
5 1−p3T 1 - p^{3} T
7 1−p3T 1 - p^{3} T
good3 1+31pT+p7T2 1 + 31 p T + p^{7} T^{2}
11 1+197pT+p7T2 1 + 197 p T + p^{7} T^{2}
13 1+1661T+p7T2 1 + 1661 T + p^{7} T^{2}
17 1+35771T+p7T2 1 + 35771 T + p^{7} T^{2}
19 1−20222T+p7T2 1 - 20222 T + p^{7} T^{2}
23 1+42130T+p7T2 1 + 42130 T + p^{7} T^{2}
29 1+111789T+p7T2 1 + 111789 T + p^{7} T^{2}
31 1+269504T+p7T2 1 + 269504 T + p^{7} T^{2}
37 1−532774T+p7T2 1 - 532774 T + p^{7} T^{2}
41 1−158056T+p7T2 1 - 158056 T + p^{7} T^{2}
43 1+521874T+p7T2 1 + 521874 T + p^{7} T^{2}
47 1+939733T+p7T2 1 + 939733 T + p^{7} T^{2}
53 1+408384T+p7T2 1 + 408384 T + p^{7} T^{2}
59 1+522172T+p7T2 1 + 522172 T + p^{7} T^{2}
61 1−350080T+p7T2 1 - 350080 T + p^{7} T^{2}
67 1+3931176T+p7T2 1 + 3931176 T + p^{7} T^{2}
71 1−1194016T+p7T2 1 - 1194016 T + p^{7} T^{2}
73 1−998350T+p7T2 1 - 998350 T + p^{7} T^{2}
79 1+2120709T+p7T2 1 + 2120709 T + p^{7} T^{2}
83 1+1746708T+p7T2 1 + 1746708 T + p^{7} T^{2}
89 1+10077740T+p7T2 1 + 10077740 T + p^{7} T^{2}
97 1+6238295T+p7T2 1 + 6238295 T + p^{7} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.69362413585645205424718153233, −11.45434015453331570780147383664, −10.96734454157312658203098730151, −9.739758237592853792796587117015, −7.37328762833955662740195748137, −6.25627768659347379788030877787, −5.33031194131864932264044013813, −4.36674476943709959838801653601, −1.77191670656768194676704463876, 0, 1.77191670656768194676704463876, 4.36674476943709959838801653601, 5.33031194131864932264044013813, 6.25627768659347379788030877787, 7.37328762833955662740195748137, 9.739758237592853792796587117015, 10.96734454157312658203098730151, 11.45434015453331570780147383664, 12.69362413585645205424718153233

Graph of the ZZ-function along the critical line