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Properties

Label 2-74-37.10-c7-0-13
Degree $2$
Conductor $74$
Sign $0.476 + 0.879i$
Analytic cond. $23.1164$
Root an. cond. $4.80796$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 6.92i)2-s + (27.4 + 47.5i)3-s + (−31.9 − 55.4i)4-s + (8.21 + 14.2i)5-s + 439.·6-s + (−392. − 679. i)7-s − 511.·8-s + (−412. + 713. i)9-s + 131.·10-s + 6.02e3·11-s + (1.75e3 − 3.04e3i)12-s + (−6.90e3 − 1.19e4i)13-s − 6.28e3·14-s + (−450. + 781. i)15-s + (−2.04e3 + 3.54e3i)16-s + (6.83e3 − 1.18e4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.586 + 1.01i)3-s + (−0.249 − 0.433i)4-s + (0.0294 + 0.0509i)5-s + 0.829·6-s + (−0.432 − 0.749i)7-s − 0.353·8-s + (−0.188 + 0.326i)9-s + 0.0415·10-s + 1.36·11-s + (0.293 − 0.508i)12-s + (−0.872 − 1.51i)13-s − 0.611·14-s + (−0.0345 + 0.0597i)15-s + (−0.125 + 0.216i)16-s + (0.337 − 0.584i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.476 + 0.879i$
Analytic conductor: \(23.1164\)
Root analytic conductor: \(4.80796\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :7/2),\ 0.476 + 0.879i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.32479 - 1.38452i\)
\(L(\frac12)\) \(\approx\) \(2.32479 - 1.38452i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 + 6.92i)T \)
37 \( 1 + (1.05e5 + 2.89e5i)T \)
good3 \( 1 + (-27.4 - 47.5i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-8.21 - 14.2i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (392. + 679. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 6.02e3T + 1.94e7T^{2} \)
13 \( 1 + (6.90e3 + 1.19e4i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-6.83e3 + 1.18e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-2.00e4 - 3.46e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 - 7.04e4T + 3.40e9T^{2} \)
29 \( 1 - 1.94e5T + 1.72e10T^{2} \)
31 \( 1 + 2.34e5T + 2.75e10T^{2} \)
41 \( 1 + (1.18e5 + 2.05e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + 8.33e5T + 2.71e11T^{2} \)
47 \( 1 + 1.45e5T + 5.06e11T^{2} \)
53 \( 1 + (-5.42e5 + 9.40e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.74e5 + 3.02e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-1.01e6 - 1.76e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.08e6 - 1.88e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-1.65e6 - 2.87e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 6.03e6T + 1.10e13T^{2} \)
79 \( 1 + (1.59e6 + 2.76e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (4.32e6 - 7.48e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (-5.00e6 + 8.66e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.05e7T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90607023130519995301399565422, −11.87264347584420175303701326718, −10.31612018141555910341550480890, −9.939138340100285604684442473839, −8.731073428067965931674959884640, −7.01307188014914454903071901276, −5.20250502720281223881223604658, −3.85793815915789452736238181697, −3.05056615658713228902102674716, −0.853656670543043251150904843106, 1.51965057226700115378126088352, 3.04582672799227104412149028817, 4.85133164697781927387077920642, 6.61281999392635242634230364253, 7.10768899868243309937872561407, 8.698116354476087081192345411931, 9.366144160792028117041069351232, 11.62238910873377996261121693283, 12.43613169729054438852588780212, 13.43283785243265705569541900226

Graph of the $Z$-function along the critical line