Properties

Label 2-38-19.11-c3-0-3
Degree $2$
Conductor $38$
Sign $0.411 + 0.911i$
Analytic cond. $2.24207$
Root an. cond. $1.49735$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−0.881 + 1.52i)3-s + (−1.99 − 3.46i)4-s + (10.3 − 17.9i)5-s + (1.76 + 3.05i)6-s + 8.76·7-s − 7.99·8-s + (11.9 + 20.6i)9-s + (−20.7 − 35.8i)10-s − 62.1·11-s + 7.05·12-s + (32.2 + 55.8i)13-s + (8.76 − 15.1i)14-s + (18.2 + 31.6i)15-s + (−8 + 13.8i)16-s + (23.2 − 40.1i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.169 + 0.293i)3-s + (−0.249 − 0.433i)4-s + (0.926 − 1.60i)5-s + (0.119 + 0.207i)6-s + 0.473·7-s − 0.353·8-s + (0.442 + 0.766i)9-s + (−0.654 − 1.13i)10-s − 1.70·11-s + 0.169·12-s + (0.688 + 1.19i)13-s + (0.167 − 0.289i)14-s + (0.314 + 0.544i)15-s + (−0.125 + 0.216i)16-s + (0.331 − 0.573i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(2.24207\)
Root analytic conductor: \(1.49735\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ 0.411 + 0.911i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.31750 - 0.850548i\)
\(L(\frac12)\) \(\approx\) \(1.31750 - 0.850548i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
19 \( 1 + (-17.3 - 80.9i)T \)
good3 \( 1 + (0.881 - 1.52i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-10.3 + 17.9i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 - 8.76T + 343T^{2} \)
11 \( 1 + 62.1T + 1.33e3T^{2} \)
13 \( 1 + (-32.2 - 55.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-23.2 + 40.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (-18.6 - 32.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-33.2 - 57.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 112.T + 2.97e4T^{2} \)
37 \( 1 + 189.T + 5.06e4T^{2} \)
41 \( 1 + (-120. + 207. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (84.1 - 145. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (93.9 + 162. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-56.5 - 97.9i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (92.9 - 161. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (154. + 267. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (19.7 + 34.1i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-175. + 303. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-4.80 + 8.31i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-588. + 1.01e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 257.T + 5.71e5T^{2} \)
89 \( 1 + (-66.8 - 115. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (598. - 1.03e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−15.90761896391524638032571124230, −13.88608641948900054418137042037, −13.27954023597370381722601666182, −12.14524713577018888618820133529, −10.63632447062895196870278951491, −9.553689601736813181482074565088, −8.187730397634418816298634978760, −5.48328212848370159723858914898, −4.68456240193701497230411617854, −1.73580625877797168425218327165, 2.95712262850221085310557982101, 5.55263532297868290397237738278, 6.69004109541386201477773036371, 7.940620693062509075743924284517, 10.04097441762633078876252909871, 11.00106171221307013276146908729, 12.86882062588539855884473084781, 13.71246487620706806879620604420, 15.00029558699280777876528418103, 15.57274662749245185744281746348

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