Properties

Degree $2$
Conductor $38$
Sign $0.136 - 0.990i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.06 − 2.57i)2-s + (−24.7 − 9.01i)3-s + (2.77 − 15.7i)4-s + (14.6 + 83.2i)5-s + (−99.1 + 36.0i)6-s + (−7.63 + 13.2i)7-s + (−32.0 − 55.4i)8-s + (346. + 290. i)9-s + (258. + 217. i)10-s + (225. + 390. i)11-s + (−210. + 365. i)12-s + (−754. + 274. i)13-s + (10.5 + 60.1i)14-s + (386. − 2.19e3i)15-s + (−240. − 87.5i)16-s + (−1.01e3 + 855. i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−1.58 − 0.578i)3-s + (0.0868 − 0.492i)4-s + (0.262 + 1.48i)5-s + (−1.12 + 0.409i)6-s + (−0.0588 + 0.101i)7-s + (−0.176 − 0.306i)8-s + (1.42 + 1.19i)9-s + (0.819 + 0.687i)10-s + (0.561 + 0.971i)11-s + (−0.422 + 0.732i)12-s + (−1.23 + 0.450i)13-s + (0.0144 + 0.0819i)14-s + (0.444 − 2.51i)15-s + (−0.234 − 0.0855i)16-s + (−0.855 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.136 - 0.990i$
Motivic weight: \(5\)
Character: $\chi_{38} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 0.136 - 0.990i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.588287 + 0.512851i\)
\(L(\frac12)\) \(\approx\) \(0.588287 + 0.512851i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.06 + 2.57i)T \)
19 \( 1 + (1.50e3 + 459. i)T \)
good3 \( 1 + (24.7 + 9.01i)T + (186. + 156. i)T^{2} \)
5 \( 1 + (-14.6 - 83.2i)T + (-2.93e3 + 1.06e3i)T^{2} \)
7 \( 1 + (7.63 - 13.2i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-225. - 390. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (754. - 274. i)T + (2.84e5 - 2.38e5i)T^{2} \)
17 \( 1 + (1.01e3 - 855. i)T + (2.46e5 - 1.39e6i)T^{2} \)
23 \( 1 + (-65.9 + 373. i)T + (-6.04e6 - 2.20e6i)T^{2} \)
29 \( 1 + (-6.38e3 - 5.35e3i)T + (3.56e6 + 2.01e7i)T^{2} \)
31 \( 1 + (-3.34e3 + 5.79e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 4.85e3T + 6.93e7T^{2} \)
41 \( 1 + (1.18e4 + 4.30e3i)T + (8.87e7 + 7.44e7i)T^{2} \)
43 \( 1 + (-2.55e3 - 1.45e4i)T + (-1.38e8 + 5.02e7i)T^{2} \)
47 \( 1 + (9.45e3 + 7.92e3i)T + (3.98e7 + 2.25e8i)T^{2} \)
53 \( 1 + (-286. + 1.62e3i)T + (-3.92e8 - 1.43e8i)T^{2} \)
59 \( 1 + (-4.24e3 + 3.55e3i)T + (1.24e8 - 7.04e8i)T^{2} \)
61 \( 1 + (-5.83e3 + 3.31e4i)T + (-7.93e8 - 2.88e8i)T^{2} \)
67 \( 1 + (-824. - 691. i)T + (2.34e8 + 1.32e9i)T^{2} \)
71 \( 1 + (-4.28e3 - 2.42e4i)T + (-1.69e9 + 6.17e8i)T^{2} \)
73 \( 1 + (-4.67e4 - 1.70e4i)T + (1.58e9 + 1.33e9i)T^{2} \)
79 \( 1 + (2.89e4 + 1.05e4i)T + (2.35e9 + 1.97e9i)T^{2} \)
83 \( 1 + (1.88e4 - 3.25e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-8.71e4 + 3.17e4i)T + (4.27e9 - 3.58e9i)T^{2} \)
97 \( 1 + (8.78e4 - 7.37e4i)T + (1.49e9 - 8.45e9i)T^{2} \)
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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

−15.31759380163255995715695640221, −14.33448749049354330896140947097, −12.80340514078238823927251138414, −11.90535475653944478446556014140, −10.90944112396902150549610708413, −10.04286037450898769593208720836, −6.89976819340604751811454783918, −6.44240924159415283368875183907, −4.67826582885240886180305684776, −2.13646882255354498416696226385, 0.44145639808686111498609112641, 4.48476188896600499318283939100, 5.28475417705690817684029230914, 6.51648878261310160418284438603, 8.672560494218737899361745267430, 10.17150156461963839842113773090, 11.72784761793478615161570717778, 12.40804618436842582355030304218, 13.65976167669869281400378221855, 15.45722533613168473495995629542

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