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Properties

Label 2-74-37.10-c7-0-11
Degree $2$
Conductor $74$
Sign $0.0760 + 0.997i$
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 + (8.02 + 13.9i)3-s + (−31.9 − 55.4i)4-s + (−60.7 − 105. i)5-s + 128.·6-s + (462. + 801. i)7-s − 511.·8-s + (964. − 1.67e3i)9-s − 972.·10-s + 1.63e3·11-s + (513. − 890. i)12-s + (1.56e3 + 2.71e3i)13-s + 7.40e3·14-s + (975. − 1.69e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (8.23e3 − 1.42e4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.171 + 0.297i)3-s + (−0.249 − 0.433i)4-s + (−0.217 − 0.376i)5-s + 0.242·6-s + (0.509 + 0.883i)7-s − 0.353·8-s + (0.441 − 0.763i)9-s − 0.307·10-s + 0.369·11-s + (0.0858 − 0.148i)12-s + (0.197 + 0.342i)13-s + 0.721·14-s + (0.0746 − 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.406 − 0.704i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0760 + 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0760 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.0760 + 0.997i$
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: \((2,\ 74,\ (\ :7/2),\ 0.0760 + 0.997i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.79071 - 1.65932i\)
\(L(\frac12)\) \(\approx\) \(1.79071 - 1.65932i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-2.18e4 + 3.07e5i)T \)
good3 \( 1 + (-8.02 - 13.9i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (60.7 + 105. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-462. - 801. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 1.63e3T + 1.94e7T^{2} \)
13 \( 1 + (-1.56e3 - 2.71e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-8.23e3 + 1.42e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (2.54e4 + 4.41e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 - 6.04e4T + 3.40e9T^{2} \)
29 \( 1 + 9.97e4T + 1.72e10T^{2} \)
31 \( 1 - 1.44e5T + 2.75e10T^{2} \)
41 \( 1 + (4.04e5 + 7.00e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + 7.00e4T + 2.71e11T^{2} \)
47 \( 1 - 7.04e5T + 5.06e11T^{2} \)
53 \( 1 + (5.73e5 - 9.93e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (4.76e5 - 8.25e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-6.70e5 - 1.16e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (1.01e6 + 1.75e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-2.52e6 - 4.37e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 - 4.62e6T + 1.10e13T^{2} \)
79 \( 1 + (-2.53e6 - 4.39e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (6.51e5 - 1.12e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (6.04e6 - 1.04e7i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 3.50e6T + 8.07e13T^{2} \)
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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

−12.62296287078764406833572777515, −11.88914477434364765422311055413, −10.83540009075327999076832236972, −9.312799961865092770033109283688, −8.740067906462451696788841134894, −6.80575391268906746177841284818, −5.19548401117202969505329463247, −4.07944344364412496133671769682, −2.51286261732955357907021711219, −0.822852523879441203252774627729, 1.45448838406535358379585285857, 3.52313703828169779567885446959, 4.80337884519493292068749385033, 6.41961363676768307658843964211, 7.56984192011830577051082987372, 8.302042095644162646701019478083, 10.15642305847020376768311871654, 11.14195540636219631439488757461, 12.63053086042851861937344700366, 13.50074370807666433401864032363

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