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: $\chi_{38} (9, \cdot )$
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

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

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