Properties

Degree $2$
Conductor $38$
Sign $0.658 - 0.752i$
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 + (−1.36 − 7.74i)3-s + (12.2 + 10.2i)4-s + (−39.1 + 32.8i)5-s + (−5.46 + 30.9i)6-s + (−9.18 + 15.9i)7-s + (−32.0 − 55.4i)8-s + (170. − 61.9i)9-s + (192. − 69.9i)10-s + (375. + 649. i)11-s + (62.9 − 108. i)12-s + (−90.9 + 515. i)13-s + (56.2 − 47.2i)14-s + (308. + 258. i)15-s + (44.4 + 252. i)16-s + (1.76e3 + 642. i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.0876 − 0.496i)3-s + (0.383 + 0.321i)4-s + (−0.700 + 0.588i)5-s + (−0.0619 + 0.351i)6-s + (−0.0708 + 0.122i)7-s + (−0.176 − 0.306i)8-s + (0.700 − 0.254i)9-s + (0.608 − 0.221i)10-s + (0.934 + 1.61i)11-s + (0.126 − 0.218i)12-s + (−0.149 + 0.846i)13-s + (0.0767 − 0.0644i)14-s + (0.353 + 0.296i)15-s + (0.0434 + 0.246i)16-s + (1.48 + 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.658 - 0.752i$
Motivic weight: \(5\)
Character: $\chi_{38} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.858030 + 0.389257i\)
\(L(\frac12)\) \(\approx\) \(0.858030 + 0.389257i\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.48e3 + 511. i)T \)
good3 \( 1 + (1.36 + 7.74i)T + (-228. + 83.1i)T^{2} \)
5 \( 1 + (39.1 - 32.8i)T + (542. - 3.07e3i)T^{2} \)
7 \( 1 + (9.18 - 15.9i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-375. - 649. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (90.9 - 515. i)T + (-3.48e5 - 1.26e5i)T^{2} \)
17 \( 1 + (-1.76e3 - 642. i)T + (1.08e6 + 9.12e5i)T^{2} \)
23 \( 1 + (374. + 314. i)T + (1.11e6 + 6.33e6i)T^{2} \)
29 \( 1 + (1.22e3 - 447. i)T + (1.57e7 - 1.31e7i)T^{2} \)
31 \( 1 + (3.27e3 - 5.67e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 8.29e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.49e3 - 8.49e3i)T + (-1.08e8 + 3.96e7i)T^{2} \)
43 \( 1 + (-1.07e4 + 8.98e3i)T + (2.55e7 - 1.44e8i)T^{2} \)
47 \( 1 + (-1.55e4 + 5.64e3i)T + (1.75e8 - 1.47e8i)T^{2} \)
53 \( 1 + (6.62e3 + 5.56e3i)T + (7.26e7 + 4.11e8i)T^{2} \)
59 \( 1 + (-1.52e4 - 5.55e3i)T + (5.47e8 + 4.59e8i)T^{2} \)
61 \( 1 + (-683. - 573. i)T + (1.46e8 + 8.31e8i)T^{2} \)
67 \( 1 + (1.67e4 - 6.08e3i)T + (1.03e9 - 8.67e8i)T^{2} \)
71 \( 1 + (-3.64e4 + 3.05e4i)T + (3.13e8 - 1.77e9i)T^{2} \)
73 \( 1 + (3.67e3 + 2.08e4i)T + (-1.94e9 + 7.09e8i)T^{2} \)
79 \( 1 + (1.47e4 + 8.39e4i)T + (-2.89e9 + 1.05e9i)T^{2} \)
83 \( 1 + (2.23e4 - 3.87e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (1.55e4 - 8.83e4i)T + (-5.24e9 - 1.90e9i)T^{2} \)
97 \( 1 + (-5.51e4 - 2.00e4i)T + (6.57e9 + 5.51e9i)T^{2} \)
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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

−15.41389002150521122209259453730, −14.51730762200772595309830454192, −12.51696299300993719620449237305, −11.97986698825724463962739050390, −10.45418102069044787211713208560, −9.247796856607374333084461246296, −7.49083945660752654770121498805, −6.75780052201399912259158872731, −3.97756635991998075767964595177, −1.68571817940766294943066243175, 0.73522371756780933252567367920, 3.81688097181680009087108191771, 5.70136467525188522980429768804, 7.60108373407104590735722356399, 8.723030939720724232212783499256, 10.06248815881345235297089924983, 11.25114629757167418010530856853, 12.54087570640048240467621703293, 14.18355756831950845399869238028, 15.52309614904198999735062474354

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