Properties

Degree $2$
Conductor $38$
Sign $0.949 + 0.314i$
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 + (18.6 − 6.79i)3-s + (2.77 + 15.7i)4-s + (18.3 − 104. i)5-s + (74.6 + 27.1i)6-s + (−18.2 − 31.5i)7-s + (−32.0 + 55.4i)8-s + (116. − 97.6i)9-s + (323. − 271. i)10-s + (−257. + 445. i)11-s + (158. + 275. i)12-s + (295. + 107. i)13-s + (25.3 − 143. i)14-s + (−364. − 2.06e3i)15-s + (−240. + 87.5i)16-s + (1.75e3 + 1.46e3i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (1.19 − 0.436i)3-s + (0.0868 + 0.492i)4-s + (0.328 − 1.86i)5-s + (0.847 + 0.308i)6-s + (−0.140 − 0.243i)7-s + (−0.176 + 0.306i)8-s + (0.478 − 0.401i)9-s + (1.02 − 0.858i)10-s + (−0.641 + 1.11i)11-s + (0.318 + 0.552i)12-s + (0.484 + 0.176i)13-s + (0.0345 − 0.195i)14-s + (−0.418 − 2.37i)15-s + (−0.234 + 0.0855i)16-s + (1.46 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.949 + 0.314i$
Motivic weight: \(5\)
Character: $\chi_{38} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 0.949 + 0.314i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.90256 - 0.468583i\)
\(L(\frac12)\) \(\approx\) \(2.90256 - 0.468583i\)
\(L(\frac{7}{2})\) 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 \( 1 + (251. - 1.55e3i)T \)
good3 \( 1 + (-18.6 + 6.79i)T + (186. - 156. i)T^{2} \)
5 \( 1 + (-18.3 + 104. i)T + (-2.93e3 - 1.06e3i)T^{2} \)
7 \( 1 + (18.2 + 31.5i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (257. - 445. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-295. - 107. i)T + (2.84e5 + 2.38e5i)T^{2} \)
17 \( 1 + (-1.75e3 - 1.46e3i)T + (2.46e5 + 1.39e6i)T^{2} \)
23 \( 1 + (350. + 1.98e3i)T + (-6.04e6 + 2.20e6i)T^{2} \)
29 \( 1 + (854. - 716. i)T + (3.56e6 - 2.01e7i)T^{2} \)
31 \( 1 + (2.58e3 + 4.48e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 4.29e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.99e3 + 727. i)T + (8.87e7 - 7.44e7i)T^{2} \)
43 \( 1 + (-1.17e3 + 6.68e3i)T + (-1.38e8 - 5.02e7i)T^{2} \)
47 \( 1 + (2.00e3 - 1.68e3i)T + (3.98e7 - 2.25e8i)T^{2} \)
53 \( 1 + (-358. - 2.03e3i)T + (-3.92e8 + 1.43e8i)T^{2} \)
59 \( 1 + (2.35e4 + 1.97e4i)T + (1.24e8 + 7.04e8i)T^{2} \)
61 \( 1 + (4.24e3 + 2.40e4i)T + (-7.93e8 + 2.88e8i)T^{2} \)
67 \( 1 + (1.54e4 - 1.29e4i)T + (2.34e8 - 1.32e9i)T^{2} \)
71 \( 1 + (1.04e4 - 5.92e4i)T + (-1.69e9 - 6.17e8i)T^{2} \)
73 \( 1 + (-3.84e4 + 1.39e4i)T + (1.58e9 - 1.33e9i)T^{2} \)
79 \( 1 + (7.76e4 - 2.82e4i)T + (2.35e9 - 1.97e9i)T^{2} \)
83 \( 1 + (-2.15e4 - 3.73e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (8.94e4 + 3.25e4i)T + (4.27e9 + 3.58e9i)T^{2} \)
97 \( 1 + (-7.95e4 - 6.67e4i)T + (1.49e9 + 8.45e9i)T^{2} \)
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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

−15.01904645407214642715487214092, −13.95141596303263529796476682706, −12.82623331118911557675415638953, −12.51523228989596649009556984218, −9.825595173017000038439313211813, −8.491476061855665533095186530620, −7.77810957408623513056194867184, −5.62204230995754193136587840848, −4.04643017784869885489238252047, −1.75233951222987119927881956637, 2.80821262071275489450421084088, 3.28268761568012469606597720376, 5.86325565803428149864511654131, 7.55762358362474914063756661562, 9.334157974330127948857786602284, 10.45834213281799394892789543816, 11.46657814312729446381684712130, 13.54951708932033355173226012260, 14.09090912236673816821741944056, 14.99506317933348673775630785495

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