Properties

Label 2-38-19.16-c7-0-2
Degree $2$
Conductor $38$
Sign $-0.632 + 0.774i$
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  + (−6.12 − 5.14i)2-s + (−64.7 + 23.5i)3-s + (11.1 + 63.0i)4-s + (−74.8 + 424. i)5-s + (517. + 188. i)6-s + (216. + 375. i)7-s + (256. − 443. i)8-s + (1.96e3 − 1.64e3i)9-s + (2.64e3 − 2.21e3i)10-s + (−3.67e3 + 6.37e3i)11-s + (−2.20e3 − 3.81e3i)12-s + (−7.03e3 − 2.56e3i)13-s + (602. − 3.41e3i)14-s + (−5.15e3 − 2.92e4i)15-s + (−3.84e3 + 1.40e3i)16-s + (1.83e4 + 1.54e4i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−1.38 + 0.503i)3-s + (0.0868 + 0.492i)4-s + (−0.267 + 1.51i)5-s + (0.978 + 0.356i)6-s + (0.238 + 0.413i)7-s + (0.176 − 0.306i)8-s + (0.896 − 0.752i)9-s + (0.835 − 0.701i)10-s + (−0.833 + 1.44i)11-s + (−0.368 − 0.637i)12-s + (−0.888 − 0.323i)13-s + (0.0586 − 0.332i)14-s + (−0.394 − 2.23i)15-s + (−0.234 + 0.0855i)16-s + (0.908 + 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.632 + 0.774i$
Analytic conductor: \(11.8706\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ -0.632 + 0.774i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0850523 - 0.179394i\)
\(L(\frac12)\) \(\approx\) \(0.0850523 - 0.179394i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.12 + 5.14i)T \)
19 \( 1 + (2.58e4 + 1.50e4i)T \)
good3 \( 1 + (64.7 - 23.5i)T + (1.67e3 - 1.40e3i)T^{2} \)
5 \( 1 + (74.8 - 424. i)T + (-7.34e4 - 2.67e4i)T^{2} \)
7 \( 1 + (-216. - 375. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (3.67e3 - 6.37e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (7.03e3 + 2.56e3i)T + (4.80e7 + 4.03e7i)T^{2} \)
17 \( 1 + (-1.83e4 - 1.54e4i)T + (7.12e7 + 4.04e8i)T^{2} \)
23 \( 1 + (5.36e3 + 3.04e4i)T + (-3.19e9 + 1.16e9i)T^{2} \)
29 \( 1 + (-1.38e5 + 1.16e5i)T + (2.99e9 - 1.69e10i)T^{2} \)
31 \( 1 + (-4.24e4 - 7.36e4i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 - 1.66e5T + 9.49e10T^{2} \)
41 \( 1 + (3.02e5 - 1.10e5i)T + (1.49e11 - 1.25e11i)T^{2} \)
43 \( 1 + (1.60e5 - 9.12e5i)T + (-2.55e11 - 9.29e10i)T^{2} \)
47 \( 1 + (-5.04e5 + 4.23e5i)T + (8.79e10 - 4.98e11i)T^{2} \)
53 \( 1 + (-1.82e5 - 1.03e6i)T + (-1.10e12 + 4.01e11i)T^{2} \)
59 \( 1 + (1.35e6 + 1.13e6i)T + (4.32e11 + 2.45e12i)T^{2} \)
61 \( 1 + (1.15e5 + 6.53e5i)T + (-2.95e12 + 1.07e12i)T^{2} \)
67 \( 1 + (3.09e6 - 2.59e6i)T + (1.05e12 - 5.96e12i)T^{2} \)
71 \( 1 + (-6.31e5 + 3.58e6i)T + (-8.54e12 - 3.11e12i)T^{2} \)
73 \( 1 + (-2.07e6 + 7.56e5i)T + (8.46e12 - 7.10e12i)T^{2} \)
79 \( 1 + (4.99e6 - 1.81e6i)T + (1.47e13 - 1.23e13i)T^{2} \)
83 \( 1 + (1.90e6 + 3.30e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (-3.97e6 - 1.44e6i)T + (3.38e13 + 2.84e13i)T^{2} \)
97 \( 1 + (-5.22e6 - 4.38e6i)T + (1.40e13 + 7.95e13i)T^{2} \)
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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

−15.50236717151578020037220817342, −14.81626858548879115194467725270, −12.53030161662757745823315622868, −11.66311725813160181163243032220, −10.42120864584502942417331090124, −10.15567292555663041312585853351, −7.72120139008493408028352323705, −6.41195035013146831596638554498, −4.67924839117792058908993303819, −2.55404643761001236001784744000, 0.15382318568382686858032387101, 1.03488147260848857232625633965, 4.90092157647098630324481790705, 5.79975148132858245845876967712, 7.48209059749566842215197795062, 8.667104205777217294874936155503, 10.36895758374458126360050572513, 11.65416145190070628154690042451, 12.55380613970636814604231407629, 13.85868017200551333959315687513

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