Properties

Degree $2$
Conductor $38$
Sign $-0.996 + 0.0860i$
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 + (7.96 − 6.68i)3-s + (−15.0 + 5.47i)4-s + (−89.7 − 32.6i)5-s + (31.8 + 26.7i)6-s + (−116. + 201. i)7-s + (−32 − 55.4i)8-s + (−23.4 + 132. i)9-s + (66.3 − 376. i)10-s + (−172. − 299. i)11-s + (−83.2 + 144. i)12-s + (387. + 325. i)13-s + (−872. − 317. i)14-s + (−933. + 339. i)15-s + (196. − 164. i)16-s + (−138. − 784. i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.511 − 0.428i)3-s + (−0.469 + 0.171i)4-s + (−1.60 − 0.584i)5-s + (0.361 + 0.303i)6-s + (−0.895 + 1.55i)7-s + (−0.176 − 0.306i)8-s + (−0.0963 + 0.546i)9-s + (0.209 − 1.18i)10-s + (−0.430 − 0.746i)11-s + (−0.166 + 0.288i)12-s + (0.636 + 0.534i)13-s + (−1.19 − 0.433i)14-s + (−1.07 + 0.390i)15-s + (0.191 − 0.160i)16-s + (−0.116 − 0.658i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0860i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.996 + 0.0860i$
Motivic weight: \(5\)
Character: $\chi_{38} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.996 + 0.0860i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0168227 - 0.390173i\)
\(L(\frac12)\) \(\approx\) \(0.0168227 - 0.390173i\)
\(L(\frac{7}{2})\) 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 \( 1 + (880. + 1.30e3i)T \)
good3 \( 1 + (-7.96 + 6.68i)T + (42.1 - 239. i)T^{2} \)
5 \( 1 + (89.7 + 32.6i)T + (2.39e3 + 2.00e3i)T^{2} \)
7 \( 1 + (116. - 201. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (172. + 299. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-387. - 325. i)T + (6.44e4 + 3.65e5i)T^{2} \)
17 \( 1 + (138. + 784. i)T + (-1.33e6 + 4.85e5i)T^{2} \)
23 \( 1 + (-1.11e3 + 407. i)T + (4.93e6 - 4.13e6i)T^{2} \)
29 \( 1 + (73.8 - 418. i)T + (-1.92e7 - 7.01e6i)T^{2} \)
31 \( 1 + (3.32e3 - 5.75e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 7.85e3T + 6.93e7T^{2} \)
41 \( 1 + (1.23e4 - 1.03e4i)T + (2.01e7 - 1.14e8i)T^{2} \)
43 \( 1 + (1.21e4 + 4.41e3i)T + (1.12e8 + 9.44e7i)T^{2} \)
47 \( 1 + (3.67e3 - 2.08e4i)T + (-2.15e8 - 7.84e7i)T^{2} \)
53 \( 1 + (-2.73e4 + 9.94e3i)T + (3.20e8 - 2.68e8i)T^{2} \)
59 \( 1 + (-1.62e3 - 9.20e3i)T + (-6.71e8 + 2.44e8i)T^{2} \)
61 \( 1 + (-7.81e3 + 2.84e3i)T + (6.46e8 - 5.42e8i)T^{2} \)
67 \( 1 + (2.49e3 - 1.41e4i)T + (-1.26e9 - 4.61e8i)T^{2} \)
71 \( 1 + (-2.93e4 - 1.06e4i)T + (1.38e9 + 1.15e9i)T^{2} \)
73 \( 1 + (-1.52e3 + 1.28e3i)T + (3.59e8 - 2.04e9i)T^{2} \)
79 \( 1 + (7.37e3 - 6.18e3i)T + (5.34e8 - 3.03e9i)T^{2} \)
83 \( 1 + (-1.70e4 + 2.96e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (8.52e4 + 7.15e4i)T + (9.69e8 + 5.49e9i)T^{2} \)
97 \( 1 + (-1.68e4 - 9.55e4i)T + (-8.06e9 + 2.93e9i)T^{2} \)
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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

−15.94131995545316852210327273072, −15.03701754966205734859000393333, −13.44893698917723044608219797648, −12.52781023453861168581751673100, −11.37586397748421341443607024638, −8.807724374862911125270534622373, −8.447438249936307293595969163022, −6.90985670202638249715253268611, −5.11373136573686452522482021205, −3.14506258521352934711349685172, 0.19851323887680771422767506815, 3.51296376658486878344653779001, 3.95020936848989029498117147904, 6.94481945451240204236396111111, 8.266369053329343715485251291399, 10.05288165705484501747363169432, 10.79498726298343194051420534487, 12.22329578099164960172265797599, 13.37408385679752733751681831384, 14.83329113562861803582256799446

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