Properties

Label 2-38-19.4-c7-0-3
Degree $2$
Conductor $38$
Sign $-0.928 - 0.372i$
Analytic cond. $11.8706$
Root an. cond. $3.44537$
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  + (7.51 + 2.73i)2-s + (11.6 + 66.3i)3-s + (49.0 + 41.1i)4-s + (−157. + 132. i)5-s + (−93.5 + 530. i)6-s + (−83.9 + 145. i)7-s + (256. + 443. i)8-s + (−2.20e3 + 802. i)9-s + (−1.54e3 + 562. i)10-s + (−154. − 267. i)11-s + (−2.15e3 + 3.73e3i)12-s + (960. − 5.44e3i)13-s + (−1.02e3 + 863. i)14-s + (−1.05e4 − 8.89e3i)15-s + (711. + 4.03e3i)16-s + (−1.35e4 − 4.94e3i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.250 + 1.41i)3-s + (0.383 + 0.321i)4-s + (−0.563 + 0.472i)5-s + (−0.176 + 1.00i)6-s + (−0.0925 + 0.160i)7-s + (0.176 + 0.306i)8-s + (−1.00 + 0.367i)9-s + (−0.488 + 0.177i)10-s + (−0.0349 − 0.0605i)11-s + (−0.359 + 0.623i)12-s + (0.121 − 0.687i)13-s + (−0.100 + 0.0841i)14-s + (−0.810 − 0.680i)15-s + (0.0434 + 0.246i)16-s + (−0.670 − 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.372i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.928 - 0.372i$
Analytic conductor: \(11.8706\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ -0.928 - 0.372i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.424113 + 2.19478i\)
\(L(\frac12)\) \(\approx\) \(0.424113 + 2.19478i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.51 - 2.73i)T \)
19 \( 1 + (-187. - 2.98e4i)T \)
good3 \( 1 + (-11.6 - 66.3i)T + (-2.05e3 + 747. i)T^{2} \)
5 \( 1 + (157. - 132. i)T + (1.35e4 - 7.69e4i)T^{2} \)
7 \( 1 + (83.9 - 145. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (154. + 267. i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (-960. + 5.44e3i)T + (-5.89e7 - 2.14e7i)T^{2} \)
17 \( 1 + (1.35e4 + 4.94e3i)T + (3.14e8 + 2.63e8i)T^{2} \)
23 \( 1 + (-5.23e4 - 4.39e4i)T + (5.91e8 + 3.35e9i)T^{2} \)
29 \( 1 + (1.03e5 - 3.76e4i)T + (1.32e10 - 1.10e10i)T^{2} \)
31 \( 1 + (-3.29e4 + 5.70e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 3.96e5T + 9.49e10T^{2} \)
41 \( 1 + (-3.91e4 - 2.21e5i)T + (-1.83e11 + 6.66e10i)T^{2} \)
43 \( 1 + (-4.07e5 + 3.41e5i)T + (4.72e10 - 2.67e11i)T^{2} \)
47 \( 1 + (-9.02e5 + 3.28e5i)T + (3.88e11 - 3.25e11i)T^{2} \)
53 \( 1 + (3.56e5 + 2.98e5i)T + (2.03e11 + 1.15e12i)T^{2} \)
59 \( 1 + (-4.69e5 - 1.71e5i)T + (1.90e12 + 1.59e12i)T^{2} \)
61 \( 1 + (5.29e5 + 4.44e5i)T + (5.45e11 + 3.09e12i)T^{2} \)
67 \( 1 + (1.76e6 - 6.43e5i)T + (4.64e12 - 3.89e12i)T^{2} \)
71 \( 1 + (2.74e6 - 2.29e6i)T + (1.57e12 - 8.95e12i)T^{2} \)
73 \( 1 + (-4.21e5 - 2.38e6i)T + (-1.03e13 + 3.77e12i)T^{2} \)
79 \( 1 + (3.40e5 + 1.93e6i)T + (-1.80e13 + 6.56e12i)T^{2} \)
83 \( 1 + (-3.03e6 + 5.25e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (-2.14e6 + 1.21e7i)T + (-4.15e13 - 1.51e13i)T^{2} \)
97 \( 1 + (-7.22e6 - 2.63e6i)T + (6.18e13 + 5.19e13i)T^{2} \)
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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

−15.28469268371830409774953063270, −14.55683600502948000232810890029, −13.08510393117386868961331192390, −11.48859615018229766648263439397, −10.52037156051321232181122754208, −9.123208740179323437218904714222, −7.53313492805612235059365554055, −5.63040004487549546161139843408, −4.17284557521371707541024964602, −3.09551694362484458829220518920, 0.78800917912724752111431459292, 2.40278848045034294883103570453, 4.41529602365768369751643535109, 6.40544401127262535106608466574, 7.49300758576453770445416800208, 8.909813504627521347391322079553, 11.06404407021733653169305655038, 12.19048897848670181447611221168, 13.02655906204920504192813521058, 13.86054100033643320603212696464

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