Properties

Label 2-38-19.7-c1-0-2
Degree $2$
Conductor $38$
Sign $0.181 + 0.983i$
Analytic cond. $0.303431$
Root an. cond. $0.550846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−1.32 − 2.29i)3-s + (−0.499 + 0.866i)4-s + (0.822 + 1.42i)5-s + (−1.32 + 2.29i)6-s + 3.64·7-s + 0.999·8-s + (−2 + 3.46i)9-s + (0.822 − 1.42i)10-s − 4.64·11-s + 2.64·12-s + (−1 + 1.73i)13-s + (−1.82 − 3.15i)14-s + (2.17 − 3.77i)15-s + (−0.5 − 0.866i)16-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.763 − 1.32i)3-s + (−0.249 + 0.433i)4-s + (0.368 + 0.637i)5-s + (−0.540 + 0.935i)6-s + 1.37·7-s + 0.353·8-s + (−0.666 + 1.15i)9-s + (0.260 − 0.450i)10-s − 1.40·11-s + 0.763·12-s + (−0.277 + 0.480i)13-s + (−0.487 − 0.843i)14-s + (0.562 − 0.973i)15-s + (−0.125 − 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(0.303431\)
Root analytic conductor: \(0.550846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :1/2),\ 0.181 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446458 - 0.371756i\)
\(L(\frac12)\) \(\approx\) \(0.446458 - 0.371756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-1.67 - 4.02i)T \)
good3 \( 1 + (1.32 + 2.29i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.822 - 1.42i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.64T + 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.822 + 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.822 - 1.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.64T + 31T^{2} \)
37 \( 1 - 0.354T + 37T^{2} \)
41 \( 1 + (-0.145 - 0.252i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.64 + 9.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.17 - 3.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.29 + 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.96 - 6.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.468 - 0.811i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.322 - 0.559i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.35 - 2.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.854 + 1.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.85 - 3.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−16.61605391119698663914357498550, −14.60847378657151987618783501230, −13.48326804328236299192940210597, −12.32155412043741912722116531154, −11.32098951143918327540100152469, −10.38455434275363389768523277618, −8.189665066694201113063771895453, −7.15013860187141160130065522007, −5.36273208423730830468980267546, −2.03226124301764104537415046840, 4.89459253855276575057430288772, 5.37304044904848557543964142803, 7.81723990346239717137102540570, 9.217517298351656496368033783061, 10.44712860429574289879660496629, 11.34993850722000104369126884548, 13.21785361130886242349146778713, 14.82612052023811621898850770595, 15.60825776508625268379754159622, 16.62070460173004578577316904015

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