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Properties

Label 2-37-37.26-c7-0-14
Degree $2$
Conductor $37$
Sign $0.894 - 0.447i$
Analytic cond. $11.5582$
Root an. cond. $3.39974$
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  + (6.29 + 10.9i)2-s + (−9.44 + 16.3i)3-s + (−15.2 + 26.4i)4-s + (125. − 217. i)5-s − 237.·6-s + (817. − 1.41e3i)7-s + 1.22e3·8-s + (915. + 1.58e3i)9-s + 3.15e3·10-s − 3.26e3·11-s + (−288. − 499. i)12-s + (285. − 494. i)13-s + 2.05e4·14-s + (2.36e3 + 4.10e3i)15-s + (9.67e3 + 1.67e4i)16-s + (−7.43e3 − 1.28e4i)17-s + ⋯
L(s)  = 1  + (0.556 + 0.963i)2-s + (−0.201 + 0.349i)3-s + (−0.119 + 0.206i)4-s + (0.448 − 0.777i)5-s − 0.449·6-s + (0.900 − 1.56i)7-s + 0.847·8-s + (0.418 + 0.724i)9-s + 0.998·10-s − 0.740·11-s + (−0.0481 − 0.0834i)12-s + (0.0360 − 0.0623i)13-s + 2.00·14-s + (0.181 + 0.313i)15-s + (0.590 + 1.02i)16-s + (−0.366 − 0.635i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.69785 + 0.637814i\)
\(L(\frac12)\) \(\approx\) \(2.69785 + 0.637814i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.70e4 + 3.06e5i)T \)
good2 \( 1 + (-6.29 - 10.9i)T + (-64 + 110. i)T^{2} \)
3 \( 1 + (9.44 - 16.3i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-125. + 217. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-817. + 1.41e3i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + 3.26e3T + 1.94e7T^{2} \)
13 \( 1 + (-285. + 494. i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (7.43e3 + 1.28e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (2.69e3 - 4.67e3i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 - 5.99e4T + 3.40e9T^{2} \)
29 \( 1 - 1.80e5T + 1.72e10T^{2} \)
31 \( 1 - 3.48e4T + 2.75e10T^{2} \)
41 \( 1 + (3.24e5 - 5.62e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + 5.49e5T + 2.71e11T^{2} \)
47 \( 1 + 1.06e6T + 5.06e11T^{2} \)
53 \( 1 + (-4.82e5 - 8.34e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-5.82e5 - 1.00e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (3.74e5 - 6.48e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (7.12e5 - 1.23e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-2.01e6 + 3.48e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + 4.04e6T + 1.10e13T^{2} \)
79 \( 1 + (-1.54e4 + 2.66e4i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-2.14e6 - 3.71e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (-3.82e6 - 6.62e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 9.54e6T + 8.07e13T^{2} \)
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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

−14.96934550768077436888631417370, −13.68561185681060105275369039797, −13.24526461765281145613937229157, −11.01149997687687249989699899655, −10.15480360236809145312820945200, −8.090275020927735389265995269785, −7.01868866888403675555972648212, −5.10912711514366125429619517474, −4.58210145135793626816439662090, −1.28768783684331761560054379488, 1.79019030748382702310822238248, 2.93106560757810076461235180897, 5.00678409567199974699831710818, 6.62158097300788726232615886751, 8.415091150179364754226539920975, 10.20632309779809811716259365530, 11.38763777812297359013714685494, 12.23199779386509227464586702915, 13.22926652710258875628363216642, 14.63192383178960071809599931967

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