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Properties

Label 2-74-1.1-c7-0-1
Degree 22
Conductor 7474
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 31.0·3-s + 64·4-s − 544.·5-s − 248.·6-s − 1.68e3·7-s − 512·8-s − 1.22e3·9-s + 4.35e3·10-s + 5.62e3·11-s + 1.98e3·12-s + 5.22e3·13-s + 1.34e4·14-s − 1.68e4·15-s + 4.09e3·16-s − 1.25e4·17-s + 9.79e3·18-s + 1.83e4·19-s − 3.48e4·20-s − 5.22e4·21-s − 4.50e4·22-s + 4.02e4·23-s − 1.58e4·24-s + 2.18e5·25-s − 4.18e4·26-s − 1.05e5·27-s − 1.07e5·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.663·3-s + 0.5·4-s − 1.94·5-s − 0.469·6-s − 1.85·7-s − 0.353·8-s − 0.560·9-s + 1.37·10-s + 1.27·11-s + 0.331·12-s + 0.660·13-s + 1.31·14-s − 1.29·15-s + 0.250·16-s − 0.620·17-s + 0.396·18-s + 0.614·19-s − 0.974·20-s − 1.23·21-s − 0.901·22-s + 0.689·23-s − 0.234·24-s + 2.79·25-s − 0.466·26-s − 1.03·27-s − 0.928·28-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: 11
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: 00
Selberg data: (2, 74, ( :7/2), 1)(2,\ 74,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 0.66308230580.6630823058
L(12)L(\frac12) ≈\approx 0.66308230580.6630823058
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 1−31.0T+2.18e3T2 1 - 31.0T + 2.18e3T^{2}
5 1+544.T+7.81e4T2 1 + 544.T + 7.81e4T^{2}
7 1+1.68e3T+8.23e5T2 1 + 1.68e3T + 8.23e5T^{2}
11 1−5.62e3T+1.94e7T2 1 - 5.62e3T + 1.94e7T^{2}
13 1−5.22e3T+6.27e7T2 1 - 5.22e3T + 6.27e7T^{2}
17 1+1.25e4T+4.10e8T2 1 + 1.25e4T + 4.10e8T^{2}
19 1−1.83e4T+8.93e8T2 1 - 1.83e4T + 8.93e8T^{2}
23 1−4.02e4T+3.40e9T2 1 - 4.02e4T + 3.40e9T^{2}
29 1−1.87e4T+1.72e10T2 1 - 1.87e4T + 1.72e10T^{2}
31 1+1.69e5T+2.75e10T2 1 + 1.69e5T + 2.75e10T^{2}
41 1+2.64e4T+1.94e11T2 1 + 2.64e4T + 1.94e11T^{2}
43 1−4.06e5T+2.71e11T2 1 - 4.06e5T + 2.71e11T^{2}
47 1−1.75e5T+5.06e11T2 1 - 1.75e5T + 5.06e11T^{2}
53 1+1.82e6T+1.17e12T2 1 + 1.82e6T + 1.17e12T^{2}
59 1+1.24e6T+2.48e12T2 1 + 1.24e6T + 2.48e12T^{2}
61 1−3.25e6T+3.14e12T2 1 - 3.25e6T + 3.14e12T^{2}
67 1+1.66e6T+6.06e12T2 1 + 1.66e6T + 6.06e12T^{2}
71 1−2.62e5T+9.09e12T2 1 - 2.62e5T + 9.09e12T^{2}
73 1−2.68e6T+1.10e13T2 1 - 2.68e6T + 1.10e13T^{2}
79 1−3.63e5T+1.92e13T2 1 - 3.63e5T + 1.92e13T^{2}
83 1+6.71e6T+2.71e13T2 1 + 6.71e6T + 2.71e13T^{2}
89 1−1.16e7T+4.42e13T2 1 - 1.16e7T + 4.42e13T^{2}
97 1−1.63e7T+8.07e13T2 1 - 1.63e7T + 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

−12.92684685138287738842199322364, −11.87467077406118727389709149179, −10.99229314924797784236458797344, −9.294355666150829675035190826528, −8.725634353394088226981647445771, −7.43043440075244597754666099101, −6.45934762301571968344495601521, −3.77905155872665571343701753993, −3.14087168157851652116598532855, −0.56121230674389349300985486201, 0.56121230674389349300985486201, 3.14087168157851652116598532855, 3.77905155872665571343701753993, 6.45934762301571968344495601521, 7.43043440075244597754666099101, 8.725634353394088226981647445771, 9.294355666150829675035190826528, 10.99229314924797784236458797344, 11.87467077406118727389709149179, 12.92684685138287738842199322364

Graph of the ZZ-function along the critical line