Properties

Degree $2$
Conductor $38$
Sign $-0.990 + 0.134i$
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 + (4.41 + 25.0i)3-s + (12.2 + 10.2i)4-s + (−71.5 + 60.0i)5-s + (17.6 − 100. i)6-s + (91.0 − 157. i)7-s + (−32.0 − 55.4i)8-s + (−377. + 137. i)9-s + (351. − 127. i)10-s + (−198. − 344. i)11-s + (−203. + 351. i)12-s + (−119. + 680. i)13-s + (−558. + 468. i)14-s + (−1.81e3 − 1.52e3i)15-s + (44.4 + 252. i)16-s + (−1.15e3 − 421. i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.282 + 1.60i)3-s + (0.383 + 0.321i)4-s + (−1.28 + 1.07i)5-s + (0.200 − 1.13i)6-s + (0.702 − 1.21i)7-s + (−0.176 − 0.306i)8-s + (−1.55 + 0.565i)9-s + (1.11 − 0.404i)10-s + (−0.495 − 0.858i)11-s + (−0.407 + 0.705i)12-s + (−0.196 + 1.11i)13-s + (−0.761 + 0.638i)14-s + (−2.08 − 1.75i)15-s + (0.0434 + 0.246i)16-s + (−0.972 − 0.353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.990 + 0.134i$
Motivic weight: \(5\)
Character: $\chi_{38} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.990 + 0.134i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0369392 - 0.547817i\)
\(L(\frac12)\) \(\approx\) \(0.0369392 - 0.547817i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-785. - 1.36e3i)T \)
good3 \( 1 + (-4.41 - 25.0i)T + (-228. + 83.1i)T^{2} \)
5 \( 1 + (71.5 - 60.0i)T + (542. - 3.07e3i)T^{2} \)
7 \( 1 + (-91.0 + 157. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (198. + 344. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (119. - 680. i)T + (-3.48e5 - 1.26e5i)T^{2} \)
17 \( 1 + (1.15e3 + 421. i)T + (1.08e6 + 9.12e5i)T^{2} \)
23 \( 1 + (-216. - 181. i)T + (1.11e6 + 6.33e6i)T^{2} \)
29 \( 1 + (4.07e3 - 1.48e3i)T + (1.57e7 - 1.31e7i)T^{2} \)
31 \( 1 + (1.34e3 - 2.32e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 3.85e3T + 6.93e7T^{2} \)
41 \( 1 + (-3.39e3 - 1.92e4i)T + (-1.08e8 + 3.96e7i)T^{2} \)
43 \( 1 + (284. - 238. i)T + (2.55e7 - 1.44e8i)T^{2} \)
47 \( 1 + (1.27e4 - 4.62e3i)T + (1.75e8 - 1.47e8i)T^{2} \)
53 \( 1 + (-4.30e3 - 3.61e3i)T + (7.26e7 + 4.11e8i)T^{2} \)
59 \( 1 + (-2.82e4 - 1.02e4i)T + (5.47e8 + 4.59e8i)T^{2} \)
61 \( 1 + (-1.69e3 - 1.42e3i)T + (1.46e8 + 8.31e8i)T^{2} \)
67 \( 1 + (249. - 90.9i)T + (1.03e9 - 8.67e8i)T^{2} \)
71 \( 1 + (-1.70e4 + 1.43e4i)T + (3.13e8 - 1.77e9i)T^{2} \)
73 \( 1 + (-9.67e3 - 5.48e4i)T + (-1.94e9 + 7.09e8i)T^{2} \)
79 \( 1 + (9.82e3 + 5.56e4i)T + (-2.89e9 + 1.05e9i)T^{2} \)
83 \( 1 + (4.25e4 - 7.37e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-3.76e3 + 2.13e4i)T + (-5.24e9 - 1.90e9i)T^{2} \)
97 \( 1 + (1.04e5 + 3.81e4i)T + (6.57e9 + 5.51e9i)T^{2} \)
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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

−16.05413822901568666482661009044, −14.91379321931548100488633390127, −14.04895915743965120285991728817, −11.25985204965013986325514132398, −11.10762207484461834910326450345, −9.937256831823726118764289827146, −8.412801509284820722389892013391, −7.21729484063745164372987231783, −4.35963027130056289785619609867, −3.34441290498922989242804398315, 0.36574411998479145561017828968, 2.11731036099219031010878801531, 5.28770559154756537252041486266, 7.29301102253709635053351313188, 8.116183385245456185955599826303, 8.888512033334952208172976263938, 11.40306466011049597847768896106, 12.32355872036585432440478394819, 13.05375394406036130039669388319, 15.04700416368871650008378681809

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