Properties

Degree $2$
Conductor $38$
Sign $-0.0930 - 0.995i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.694 − 3.93i)2-s + (−8.20 − 6.88i)3-s + (−15.0 − 5.47i)4-s + (−55.8 + 20.3i)5-s + (−32.8 + 27.5i)6-s + (115. + 199. i)7-s + (−32 + 55.4i)8-s + (−22.2 − 126. i)9-s + (41.2 + 234. i)10-s + (28.7 − 49.7i)11-s + (85.6 + 148. i)12-s + (−825. + 692. i)13-s + (866. − 315. i)14-s + (597. + 217. i)15-s + (196. + 164. i)16-s + (−149. + 850. i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.526 − 0.441i)3-s + (−0.469 − 0.171i)4-s + (−0.998 + 0.363i)5-s + (−0.372 + 0.312i)6-s + (0.888 + 1.53i)7-s + (−0.176 + 0.306i)8-s + (−0.0916 − 0.519i)9-s + (0.130 + 0.740i)10-s + (0.0716 − 0.124i)11-s + (0.171 + 0.297i)12-s + (−1.35 + 1.13i)13-s + (1.18 − 0.429i)14-s + (0.686 + 0.249i)15-s + (0.191 + 0.160i)16-s + (−0.125 + 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0930 - 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0930 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.0930 - 0.995i$
Motivic weight: \(5\)
Character: $\chi_{38} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.0930 - 0.995i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.266575 + 0.292659i\)
\(L(\frac12)\) \(\approx\) \(0.266575 + 0.292659i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.694 + 3.93i)T \)
19 \( 1 + (1.44e3 + 633. i)T \)
good3 \( 1 + (8.20 + 6.88i)T + (42.1 + 239. i)T^{2} \)
5 \( 1 + (55.8 - 20.3i)T + (2.39e3 - 2.00e3i)T^{2} \)
7 \( 1 + (-115. - 199. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-28.7 + 49.7i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (825. - 692. i)T + (6.44e4 - 3.65e5i)T^{2} \)
17 \( 1 + (149. - 850. i)T + (-1.33e6 - 4.85e5i)T^{2} \)
23 \( 1 + (1.49e3 + 544. i)T + (4.93e6 + 4.13e6i)T^{2} \)
29 \( 1 + (1.25e3 + 7.12e3i)T + (-1.92e7 + 7.01e6i)T^{2} \)
31 \( 1 + (601. + 1.04e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 1.44e4T + 6.93e7T^{2} \)
41 \( 1 + (-1.12e4 - 9.42e3i)T + (2.01e7 + 1.14e8i)T^{2} \)
43 \( 1 + (1.63e4 - 5.93e3i)T + (1.12e8 - 9.44e7i)T^{2} \)
47 \( 1 + (785. + 4.45e3i)T + (-2.15e8 + 7.84e7i)T^{2} \)
53 \( 1 + (573. + 208. i)T + (3.20e8 + 2.68e8i)T^{2} \)
59 \( 1 + (1.88e3 - 1.07e4i)T + (-6.71e8 - 2.44e8i)T^{2} \)
61 \( 1 + (1.59e4 + 5.81e3i)T + (6.46e8 + 5.42e8i)T^{2} \)
67 \( 1 + (1.30e3 + 7.38e3i)T + (-1.26e9 + 4.61e8i)T^{2} \)
71 \( 1 + (-2.77e4 + 1.01e4i)T + (1.38e9 - 1.15e9i)T^{2} \)
73 \( 1 + (-2.47e4 - 2.07e4i)T + (3.59e8 + 2.04e9i)T^{2} \)
79 \( 1 + (2.36e4 + 1.98e4i)T + (5.34e8 + 3.03e9i)T^{2} \)
83 \( 1 + (1.87e4 + 3.24e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (6.39e3 - 5.36e3i)T + (9.69e8 - 5.49e9i)T^{2} \)
97 \( 1 + (7.30e3 - 4.14e4i)T + (-8.06e9 - 2.93e9i)T^{2} \)
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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

−15.14708791397097383289005187128, −14.70113524142166235418536858991, −12.68178490041511549805409527525, −11.69427955081152224351426093407, −11.44590244848684811428634321675, −9.373321672842540095907134468666, −7.999555889357858115905935488301, −6.17584639200331232613518005680, −4.42075957654681948227761307313, −2.23723577448634154013816981948, 0.22627110144835400123007686983, 4.20829930971042469622553525549, 5.09696406972937024038050839794, 7.37235325014014421800265156863, 8.047100564611777405093799523771, 10.15989810538609631551824781702, 11.20543003286860670789131790870, 12.60970748509347213112356727689, 14.06087927583854463736932863295, 15.07780158184546330298368182796

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