Properties

Degree $2$
Conductor $38$
Sign $0.0961 - 0.995i$
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 + (−0.196 + 1.11i)3-s + (12.2 − 10.2i)4-s + (8.74 + 7.34i)5-s + (−0.785 − 4.45i)6-s + (−7.74 − 13.4i)7-s + (−32.0 + 55.4i)8-s + (227. + 82.6i)9-s + (−42.9 − 15.6i)10-s + (−298. + 516. i)11-s + (9.04 + 15.6i)12-s + (176. + 1.00e3i)13-s + (47.4 + 39.8i)14-s + (−9.89 + 8.30i)15-s + (44.4 − 252. i)16-s + (−244. + 89.0i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.0126 + 0.0714i)3-s + (0.383 − 0.321i)4-s + (0.156 + 0.131i)5-s + (−0.00891 − 0.0505i)6-s + (−0.0597 − 0.103i)7-s + (−0.176 + 0.306i)8-s + (0.934 + 0.340i)9-s + (−0.135 − 0.0494i)10-s + (−0.743 + 1.28i)11-s + (0.0181 + 0.0314i)12-s + (0.290 + 1.64i)13-s + (0.0646 + 0.0542i)14-s + (−0.0113 + 0.00953i)15-s + (0.0434 − 0.246i)16-s + (−0.205 + 0.0747i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0961 - 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0961 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.0961 - 0.995i$
Motivic weight: \(5\)
Character: $\chi_{38} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 0.0961 - 0.995i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.815133 + 0.740188i\)
\(L(\frac12)\) \(\approx\) \(0.815133 + 0.740188i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-1.39e3 - 732. i)T \)
good3 \( 1 + (0.196 - 1.11i)T + (-228. - 83.1i)T^{2} \)
5 \( 1 + (-8.74 - 7.34i)T + (542. + 3.07e3i)T^{2} \)
7 \( 1 + (7.74 + 13.4i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (298. - 516. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-176. - 1.00e3i)T + (-3.48e5 + 1.26e5i)T^{2} \)
17 \( 1 + (244. - 89.0i)T + (1.08e6 - 9.12e5i)T^{2} \)
23 \( 1 + (-144. + 121. i)T + (1.11e6 - 6.33e6i)T^{2} \)
29 \( 1 + (-2.81e3 - 1.02e3i)T + (1.57e7 + 1.31e7i)T^{2} \)
31 \( 1 + (2.44e3 + 4.23e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 3.79e3T + 6.93e7T^{2} \)
41 \( 1 + (3.04e3 - 1.72e4i)T + (-1.08e8 - 3.96e7i)T^{2} \)
43 \( 1 + (-2.76e3 - 2.31e3i)T + (2.55e7 + 1.44e8i)T^{2} \)
47 \( 1 + (1.18e4 + 4.31e3i)T + (1.75e8 + 1.47e8i)T^{2} \)
53 \( 1 + (4.93e3 - 4.14e3i)T + (7.26e7 - 4.11e8i)T^{2} \)
59 \( 1 + (-1.38e3 + 503. i)T + (5.47e8 - 4.59e8i)T^{2} \)
61 \( 1 + (-2.66e4 + 2.23e4i)T + (1.46e8 - 8.31e8i)T^{2} \)
67 \( 1 + (4.95e4 + 1.80e4i)T + (1.03e9 + 8.67e8i)T^{2} \)
71 \( 1 + (1.98e3 + 1.66e3i)T + (3.13e8 + 1.77e9i)T^{2} \)
73 \( 1 + (-3.92e3 + 2.22e4i)T + (-1.94e9 - 7.09e8i)T^{2} \)
79 \( 1 + (-6.69e3 + 3.79e4i)T + (-2.89e9 - 1.05e9i)T^{2} \)
83 \( 1 + (4.50e4 + 7.80e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-1.55e4 - 8.81e4i)T + (-5.24e9 + 1.90e9i)T^{2} \)
97 \( 1 + (-1.26e5 + 4.58e4i)T + (6.57e9 - 5.51e9i)T^{2} \)
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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

−15.82446357561893430875088848245, −14.57544353262981091732636489171, −13.22812488023059307189235284843, −11.79355528692825952476518401998, −10.30348885688371553013216541916, −9.473121770849611110243366808604, −7.75555390569933698249561822706, −6.63637172329271231376192638757, −4.58791913299770767416599887621, −1.87415412496480074407177498487, 0.827859708001106926218566088348, 3.16473058116156958061232073104, 5.60862037675442822970544868243, 7.40595952077570680340550640499, 8.677156602569655700348902293741, 10.06559517185334544786586822181, 11.09276338660596880950714289903, 12.62259070427878073523088314002, 13.53927929786699406768850002579, 15.44660554664597100943824641329

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