Properties

Label 2-38-19.17-c3-0-3
Degree $2$
Conductor $38$
Sign $0.418 + 0.908i$
Analytic cond. $2.24207$
Root an. cond. $1.49735$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.347 + 1.96i)2-s + (−5.68 − 4.76i)3-s + (−3.75 − 1.36i)4-s + (18.4 − 6.70i)5-s + (11.3 − 9.53i)6-s + (−15.6 − 27.0i)7-s + (4 − 6.92i)8-s + (4.86 + 27.5i)9-s + (6.80 + 38.6i)10-s + (−5.32 + 9.23i)11-s + (14.8 + 25.6i)12-s + (−13.5 + 11.3i)13-s + (58.6 − 21.3i)14-s + (−136. − 49.7i)15-s + (12.2 + 10.2i)16-s + (4.26 − 24.1i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−1.09 − 0.917i)3-s + (−0.469 − 0.171i)4-s + (1.64 − 0.599i)5-s + (0.773 − 0.648i)6-s + (−0.842 − 1.45i)7-s + (0.176 − 0.306i)8-s + (0.180 + 1.02i)9-s + (0.215 + 1.22i)10-s + (−0.146 + 0.253i)11-s + (0.356 + 0.618i)12-s + (−0.288 + 0.242i)13-s + (1.11 − 0.407i)14-s + (−2.35 − 0.855i)15-s + (0.191 + 0.160i)16-s + (0.0608 − 0.345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.418 + 0.908i$
Analytic conductor: \(2.24207\)
Root analytic conductor: \(1.49735\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ 0.418 + 0.908i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.766279 - 0.490585i\)
\(L(\frac12)\) \(\approx\) \(0.766279 - 0.490585i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.347 - 1.96i)T \)
19 \( 1 + (-82.5 - 6.28i)T \)
good3 \( 1 + (5.68 + 4.76i)T + (4.68 + 26.5i)T^{2} \)
5 \( 1 + (-18.4 + 6.70i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (15.6 + 27.0i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (5.32 - 9.23i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (13.5 - 11.3i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (-4.26 + 24.1i)T + (-4.61e3 - 1.68e3i)T^{2} \)
23 \( 1 + (-46.1 - 16.8i)T + (9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (18.7 + 106. i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (-147. - 256. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 68.5T + 5.06e4T^{2} \)
41 \( 1 + (-39.8 - 33.4i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (-273. + 99.5i)T + (6.09e4 - 5.11e4i)T^{2} \)
47 \( 1 + (47.7 + 270. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + (-318. - 116. i)T + (1.14e5 + 9.56e4i)T^{2} \)
59 \( 1 + (19.1 - 108. i)T + (-1.92e5 - 7.02e4i)T^{2} \)
61 \( 1 + (401. + 146. i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (122. + 694. i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (250. - 91.0i)T + (2.74e5 - 2.30e5i)T^{2} \)
73 \( 1 + (-689. - 578. i)T + (6.75e4 + 3.83e5i)T^{2} \)
79 \( 1 + (-372. - 312. i)T + (8.56e4 + 4.85e5i)T^{2} \)
83 \( 1 + (91.6 + 158. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-303. + 254. i)T + (1.22e5 - 6.94e5i)T^{2} \)
97 \( 1 + (-119. + 675. i)T + (-8.57e5 - 3.12e5i)T^{2} \)
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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

−16.19557293953252975290704247410, −13.90707449116893636640503317345, −13.42239973055720435589324412471, −12.36980414865654730697636948035, −10.42698348296670959848612238702, −9.457113143189875099453741136463, −7.24557993376303301568311904523, −6.37274377334070495189326026644, −5.15818443770317536821666744455, −0.989173392947323134565786618148, 2.72897573199322621498354974333, 5.36257678093405555117861929700, 6.11655185780760764477062036624, 9.278562274636402578645027490934, 9.910868783855234030949152142258, 10.94602727313602148853844689806, 12.21767696557828760238420708743, 13.43157206999043903606454849177, 14.95697278739465414800971869639, 16.20851321460227309782099357704

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