Properties

Label 2-38-19.5-c7-0-5
Degree $2$
Conductor $38$
Sign $0.616 + 0.787i$
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 + (5.10 − 28.9i)3-s + (49.0 − 41.1i)4-s + (128. + 108. i)5-s + (40.8 + 231. i)6-s + (−158. − 274. i)7-s + (−256. + 443. i)8-s + (1.24e3 + 452. i)9-s + (−1.26e3 − 459. i)10-s + (−2.15e3 + 3.72e3i)11-s + (−940. − 1.62e3i)12-s + (−2.49e3 − 1.41e4i)13-s + (1.94e3 + 1.63e3i)14-s + (3.78e3 − 3.17e3i)15-s + (711. − 4.03e3i)16-s + (2.78e4 − 1.01e4i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.109 − 0.618i)3-s + (0.383 − 0.321i)4-s + (0.460 + 0.386i)5-s + (0.0771 + 0.437i)6-s + (−0.174 − 0.302i)7-s + (−0.176 + 0.306i)8-s + (0.568 + 0.207i)9-s + (−0.399 − 0.145i)10-s + (−0.487 + 0.844i)11-s + (−0.157 − 0.272i)12-s + (−0.314 − 1.78i)13-s + (0.189 + 0.158i)14-s + (0.289 − 0.242i)15-s + (0.0434 − 0.246i)16-s + (1.37 − 0.499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.616 + 0.787i$
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: \((2,\ 38,\ (\ :7/2),\ 0.616 + 0.787i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.27969 - 0.622927i\)
\(L(\frac12)\) \(\approx\) \(1.27969 - 0.622927i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-2.87e4 + 8.16e3i)T \)
good3 \( 1 + (-5.10 + 28.9i)T + (-2.05e3 - 747. i)T^{2} \)
5 \( 1 + (-128. - 108. i)T + (1.35e4 + 7.69e4i)T^{2} \)
7 \( 1 + (158. + 274. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (2.15e3 - 3.72e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (2.49e3 + 1.41e4i)T + (-5.89e7 + 2.14e7i)T^{2} \)
17 \( 1 + (-2.78e4 + 1.01e4i)T + (3.14e8 - 2.63e8i)T^{2} \)
23 \( 1 + (-5.75e4 + 4.82e4i)T + (5.91e8 - 3.35e9i)T^{2} \)
29 \( 1 + (2.27e5 + 8.29e4i)T + (1.32e10 + 1.10e10i)T^{2} \)
31 \( 1 + (-5.16e4 - 8.94e4i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + 6.74e4T + 9.49e10T^{2} \)
41 \( 1 + (3.11e4 - 1.76e5i)T + (-1.83e11 - 6.66e10i)T^{2} \)
43 \( 1 + (-9.91e3 - 8.31e3i)T + (4.72e10 + 2.67e11i)T^{2} \)
47 \( 1 + (-8.45e5 - 3.07e5i)T + (3.88e11 + 3.25e11i)T^{2} \)
53 \( 1 + (1.07e5 - 9.03e4i)T + (2.03e11 - 1.15e12i)T^{2} \)
59 \( 1 + (-1.22e6 + 4.45e5i)T + (1.90e12 - 1.59e12i)T^{2} \)
61 \( 1 + (2.08e6 - 1.74e6i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (-9.61e5 - 3.49e5i)T + (4.64e12 + 3.89e12i)T^{2} \)
71 \( 1 + (-1.90e6 - 1.59e6i)T + (1.57e12 + 8.95e12i)T^{2} \)
73 \( 1 + (6.69e5 - 3.79e6i)T + (-1.03e13 - 3.77e12i)T^{2} \)
79 \( 1 + (-6.05e5 + 3.43e6i)T + (-1.80e13 - 6.56e12i)T^{2} \)
83 \( 1 + (9.27e4 + 1.60e5i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (3.06e5 + 1.74e6i)T + (-4.15e13 + 1.51e13i)T^{2} \)
97 \( 1 + (1.20e7 - 4.39e6i)T + (6.18e13 - 5.19e13i)T^{2} \)
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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

−14.72798696742538980540945666058, −13.35634169773354424705003125504, −12.34413468595068220359077184655, −10.47507959050378959751427236263, −9.796976589660360682780873731629, −7.82995828694488480720175677343, −7.13181539712622974840695613961, −5.41163105902175989411690642754, −2.66524797104488342002545873249, −0.867623130019392639413693399078, 1.46769313095169785629891053507, 3.54978950397656073643534726350, 5.50027855417283782261059979496, 7.36062672854509284769771241494, 9.118533591271609639970284573871, 9.658501754490736088914037912317, 11.09002790253722540270100259842, 12.37523108429522497936296358891, 13.75053583466847096334968595520, 15.18264977335907213539021577306

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