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Properties

Label 2-74-1.1-c7-0-17
Degree 22
Conductor 7474
Sign −1-1
Analytic cond. 23.116423.1164
Root an. cond. 4.807964.80796
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 55.2·3-s + 64·4-s − 82.2·5-s − 441.·6-s + 195.·7-s − 512·8-s + 863.·9-s + 658.·10-s − 3.90e3·11-s + 3.53e3·12-s − 1.48e4·13-s − 1.56e3·14-s − 4.54e3·15-s + 4.09e3·16-s − 6.73e3·17-s − 6.90e3·18-s + 3.69e4·19-s − 5.26e3·20-s + 1.07e4·21-s + 3.12e4·22-s + 4.62e4·23-s − 2.82e4·24-s − 7.13e4·25-s + 1.18e5·26-s − 7.31e4·27-s + 1.25e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.18·3-s + 0.5·4-s − 0.294·5-s − 0.835·6-s + 0.215·7-s − 0.353·8-s + 0.394·9-s + 0.208·10-s − 0.883·11-s + 0.590·12-s − 1.87·13-s − 0.152·14-s − 0.347·15-s + 0.250·16-s − 0.332·17-s − 0.279·18-s + 1.23·19-s − 0.147·20-s + 0.254·21-s + 0.624·22-s + 0.793·23-s − 0.417·24-s − 0.913·25-s + 1.32·26-s − 0.714·27-s + 0.107·28-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −1-1
Analytic conductor: 23.116423.1164
Root analytic conductor: 4.807964.80796
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 74, ( :7/2), −1)(2,\ 74,\ (\ :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+8T 1 + 8T
37 1−5.06e4T 1 - 5.06e4T
good3 1−55.2T+2.18e3T2 1 - 55.2T + 2.18e3T^{2}
5 1+82.2T+7.81e4T2 1 + 82.2T + 7.81e4T^{2}
7 1−195.T+8.23e5T2 1 - 195.T + 8.23e5T^{2}
11 1+3.90e3T+1.94e7T2 1 + 3.90e3T + 1.94e7T^{2}
13 1+1.48e4T+6.27e7T2 1 + 1.48e4T + 6.27e7T^{2}
17 1+6.73e3T+4.10e8T2 1 + 6.73e3T + 4.10e8T^{2}
19 1−3.69e4T+8.93e8T2 1 - 3.69e4T + 8.93e8T^{2}
23 1−4.62e4T+3.40e9T2 1 - 4.62e4T + 3.40e9T^{2}
29 1−6.00e4T+1.72e10T2 1 - 6.00e4T + 1.72e10T^{2}
31 1+3.05e5T+2.75e10T2 1 + 3.05e5T + 2.75e10T^{2}
41 1+6.70e5T+1.94e11T2 1 + 6.70e5T + 1.94e11T^{2}
43 1−2.20e5T+2.71e11T2 1 - 2.20e5T + 2.71e11T^{2}
47 1−2.92e5T+5.06e11T2 1 - 2.92e5T + 5.06e11T^{2}
53 1+7.15e5T+1.17e12T2 1 + 7.15e5T + 1.17e12T^{2}
59 1−9.88e5T+2.48e12T2 1 - 9.88e5T + 2.48e12T^{2}
61 1+3.06e6T+3.14e12T2 1 + 3.06e6T + 3.14e12T^{2}
67 1−7.27e5T+6.06e12T2 1 - 7.27e5T + 6.06e12T^{2}
71 1−5.23e6T+9.09e12T2 1 - 5.23e6T + 9.09e12T^{2}
73 1+2.48e6T+1.10e13T2 1 + 2.48e6T + 1.10e13T^{2}
79 1−6.57e6T+1.92e13T2 1 - 6.57e6T + 1.92e13T^{2}
83 1+4.65e6T+2.71e13T2 1 + 4.65e6T + 2.71e13T^{2}
89 1−1.44e6T+4.42e13T2 1 - 1.44e6T + 4.42e13T^{2}
97 1+5.97e5T+8.07e13T2 1 + 5.97e5T + 8.07e13T^{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

−12.57131314614407240130830775474, −11.34944216406116872741069675876, −9.962508126897688723390297842405, −9.130898114351751010074685737206, −7.893191366413452225076333575503, −7.31232586276366107171470739243, −5.13599974824715060865115092151, −3.18869547090167069367072090705, −2.09273615838921581829043172705, 0, 2.09273615838921581829043172705, 3.18869547090167069367072090705, 5.13599974824715060865115092151, 7.31232586276366107171470739243, 7.893191366413452225076333575503, 9.130898114351751010074685737206, 9.962508126897688723390297842405, 11.34944216406116872741069675876, 12.57131314614407240130830775474

Graph of the ZZ-function along the critical line