Properties

Degree $2$
Conductor $38$
Sign $-0.972 + 0.234i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.75 − 1.36i)2-s + (−5.14 − 29.1i)3-s + (12.2 + 10.2i)4-s + (68.5 − 57.5i)5-s + (−20.5 + 116. i)6-s + (68.7 − 119. i)7-s + (−32.0 − 55.4i)8-s + (−597. + 217. i)9-s + (−336. + 122. i)10-s + (152. + 264. i)11-s + (237. − 410. i)12-s + (15.9 − 90.4i)13-s + (−421. + 353. i)14-s + (−2.03e3 − 1.70e3i)15-s + (44.4 + 252. i)16-s + (54.3 + 19.7i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.330 − 1.87i)3-s + (0.383 + 0.321i)4-s + (1.22 − 1.02i)5-s + (−0.233 + 1.32i)6-s + (0.529 − 0.917i)7-s + (−0.176 − 0.306i)8-s + (−2.46 + 0.895i)9-s + (−1.06 + 0.387i)10-s + (0.380 + 0.658i)11-s + (0.475 − 0.823i)12-s + (0.0261 − 0.148i)13-s + (−0.574 + 0.481i)14-s + (−2.33 − 1.95i)15-s + (0.0434 + 0.246i)16-s + (0.0455 + 0.0165i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.972 + 0.234i$
Motivic weight: \(5\)
Character: $\chi_{38} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.972 + 0.234i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.147199 - 1.23708i\)
\(L(\frac12)\) \(\approx\) \(0.147199 - 1.23708i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.75 + 1.36i)T \)
19 \( 1 + (-920. - 1.27e3i)T \)
good3 \( 1 + (5.14 + 29.1i)T + (-228. + 83.1i)T^{2} \)
5 \( 1 + (-68.5 + 57.5i)T + (542. - 3.07e3i)T^{2} \)
7 \( 1 + (-68.7 + 119. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-152. - 264. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-15.9 + 90.4i)T + (-3.48e5 - 1.26e5i)T^{2} \)
17 \( 1 + (-54.3 - 19.7i)T + (1.08e6 + 9.12e5i)T^{2} \)
23 \( 1 + (-2.22e3 - 1.86e3i)T + (1.11e6 + 6.33e6i)T^{2} \)
29 \( 1 + (5.07e3 - 1.84e3i)T + (1.57e7 - 1.31e7i)T^{2} \)
31 \( 1 + (-2.96e3 + 5.14e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 287.T + 6.93e7T^{2} \)
41 \( 1 + (240. + 1.36e3i)T + (-1.08e8 + 3.96e7i)T^{2} \)
43 \( 1 + (-6.18e3 + 5.18e3i)T + (2.55e7 - 1.44e8i)T^{2} \)
47 \( 1 + (1.47e3 - 537. i)T + (1.75e8 - 1.47e8i)T^{2} \)
53 \( 1 + (2.96e4 + 2.49e4i)T + (7.26e7 + 4.11e8i)T^{2} \)
59 \( 1 + (1.90e3 + 692. i)T + (5.47e8 + 4.59e8i)T^{2} \)
61 \( 1 + (-1.66e4 - 1.40e4i)T + (1.46e8 + 8.31e8i)T^{2} \)
67 \( 1 + (-5.93e4 + 2.15e4i)T + (1.03e9 - 8.67e8i)T^{2} \)
71 \( 1 + (1.92e4 - 1.61e4i)T + (3.13e8 - 1.77e9i)T^{2} \)
73 \( 1 + (-5.90e3 - 3.34e4i)T + (-1.94e9 + 7.09e8i)T^{2} \)
79 \( 1 + (2.42e3 + 1.37e4i)T + (-2.89e9 + 1.05e9i)T^{2} \)
83 \( 1 + (4.01e4 - 6.94e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (8.64e3 - 4.90e4i)T + (-5.24e9 - 1.90e9i)T^{2} \)
97 \( 1 + (3.82e4 + 1.39e4i)T + (6.57e9 + 5.51e9i)T^{2} \)
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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

−14.20095149949863315780407272801, −13.25660067884646484089116220287, −12.49163404167757324159871306148, −11.24970574783496713926547119602, −9.563138613879757643461870521377, −8.108325996693400991333855053770, −7.03148017442381195262131042765, −5.55552068160393133407887762196, −1.81802609134381555725091027169, −1.00519814906895539603939734219, 2.85833287872900030965558457112, 5.21067240844862921629912583258, 6.27233104352871197448671235209, 8.856765525352505689984520364295, 9.596954804556284649919876075745, 10.72701470910156754746403800522, 11.42388990574730391495470165084, 14.15240535113608796289885010775, 14.88134358110950747981735154228, 15.81951776928165121173093072282

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