Properties

Label 2-38-19.5-c7-0-3
Degree $2$
Conductor $38$
Sign $0.944 - 0.328i$
Analytic cond. $11.8706$
Root an. cond. $3.44537$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.51 − 2.73i)2-s + (−3.06 + 17.4i)3-s + (49.0 − 41.1i)4-s + (−9.97 − 8.37i)5-s + (24.5 + 139. i)6-s + (364. + 631. i)7-s + (256. − 443. i)8-s + (1.76e3 + 641. i)9-s + (−97.9 − 35.6i)10-s + (2.19e3 − 3.79e3i)11-s + (565. + 979. i)12-s + (2.25e3 + 1.27e4i)13-s + (4.47e3 + 3.75e3i)14-s + (176. − 147. i)15-s + (711. − 4.03e3i)16-s + (1.90e4 − 6.94e3i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−0.0656 + 0.372i)3-s + (0.383 − 0.321i)4-s + (−0.0356 − 0.0299i)5-s + (0.0464 + 0.263i)6-s + (0.401 + 0.696i)7-s + (0.176 − 0.306i)8-s + (0.805 + 0.293i)9-s + (−0.0309 − 0.0112i)10-s + (0.496 − 0.859i)11-s + (0.0945 + 0.163i)12-s + (0.284 + 1.61i)13-s + (0.435 + 0.365i)14-s + (0.0134 − 0.0113i)15-s + (0.0434 − 0.246i)16-s + (0.941 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(11.8706\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ 0.944 - 0.328i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.81088 + 0.474469i\)
\(L(\frac12)\) \(\approx\) \(2.81088 + 0.474469i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.51 + 2.73i)T \)
19 \( 1 + (1.60e3 - 2.98e4i)T \)
good3 \( 1 + (3.06 - 17.4i)T + (-2.05e3 - 747. i)T^{2} \)
5 \( 1 + (9.97 + 8.37i)T + (1.35e4 + 7.69e4i)T^{2} \)
7 \( 1 + (-364. - 631. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (-2.19e3 + 3.79e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (-2.25e3 - 1.27e4i)T + (-5.89e7 + 2.14e7i)T^{2} \)
17 \( 1 + (-1.90e4 + 6.94e3i)T + (3.14e8 - 2.63e8i)T^{2} \)
23 \( 1 + (-4.64e4 + 3.89e4i)T + (5.91e8 - 3.35e9i)T^{2} \)
29 \( 1 + (9.54e4 + 3.47e4i)T + (1.32e10 + 1.10e10i)T^{2} \)
31 \( 1 + (-7.62e4 - 1.32e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + 4.86e5T + 9.49e10T^{2} \)
41 \( 1 + (3.74e3 - 2.12e4i)T + (-1.83e11 - 6.66e10i)T^{2} \)
43 \( 1 + (5.38e5 + 4.51e5i)T + (4.72e10 + 2.67e11i)T^{2} \)
47 \( 1 + (9.61e5 + 3.49e5i)T + (3.88e11 + 3.25e11i)T^{2} \)
53 \( 1 + (7.88e5 - 6.61e5i)T + (2.03e11 - 1.15e12i)T^{2} \)
59 \( 1 + (-2.38e6 + 8.68e5i)T + (1.90e12 - 1.59e12i)T^{2} \)
61 \( 1 + (3.67e4 - 3.08e4i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (-4.29e6 - 1.56e6i)T + (4.64e12 + 3.89e12i)T^{2} \)
71 \( 1 + (6.73e5 + 5.64e5i)T + (1.57e12 + 8.95e12i)T^{2} \)
73 \( 1 + (-1.38e5 + 7.84e5i)T + (-1.03e13 - 3.77e12i)T^{2} \)
79 \( 1 + (-4.39e5 + 2.49e6i)T + (-1.80e13 - 6.56e12i)T^{2} \)
83 \( 1 + (2.46e5 + 4.26e5i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (5.41e5 + 3.07e6i)T + (-4.15e13 + 1.51e13i)T^{2} \)
97 \( 1 + (1.07e7 - 3.91e6i)T + (6.18e13 - 5.19e13i)T^{2} \)
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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

−14.64516888775079525028719910841, −13.81678999004966386883330442257, −12.29270595973487308321003806215, −11.41858845871844721779933682831, −10.05053717362054349837576013405, −8.596093469898054402827603820442, −6.68004579095673978756223811865, −5.14390801244430044507358085062, −3.74905613140459284802351946599, −1.69627309397656143317247877959, 1.28346777083983728620284865109, 3.57967674147947348768156145987, 5.15735477500425948267566640359, 6.88978058478167019177505419337, 7.82187743463170757362242581619, 9.868549520507272285162877935669, 11.24954555846202462761850555679, 12.65911587520577849089361516526, 13.33413054032538993803642234279, 14.81923920921539852223600877755

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