Properties

Label 2-38-19.7-c5-0-0
Degree $2$
Conductor $38$
Sign $-0.941 - 0.337i$
Analytic cond. $6.09458$
Root an. cond. $2.46872$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−13.9 − 24.1i)3-s + (−7.99 + 13.8i)4-s + (34.6 + 59.9i)5-s + (55.6 − 96.4i)6-s − 195.·7-s − 63.9·8-s + (−265. + 460. i)9-s + (−138. + 239. i)10-s − 137.·11-s + 445.·12-s + (−398. + 689. i)13-s + (−390. − 676. i)14-s + (963. − 1.66e3i)15-s + (−128 − 221. i)16-s + (−784. − 1.35e3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.892 − 1.54i)3-s + (−0.249 + 0.433i)4-s + (0.619 + 1.07i)5-s + (0.631 − 1.09i)6-s − 1.50·7-s − 0.353·8-s + (−1.09 + 1.89i)9-s + (−0.437 + 0.758i)10-s − 0.342·11-s + 0.892·12-s + (−0.653 + 1.13i)13-s + (−0.532 − 0.922i)14-s + (1.10 − 1.91i)15-s + (−0.125 − 0.216i)16-s + (−0.658 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.941 - 0.337i$
Analytic conductor: \(6.09458\)
Root analytic conductor: \(2.46872\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.941 - 0.337i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0530660 + 0.305182i\)
\(L(\frac12)\) \(\approx\) \(0.0530660 + 0.305182i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
19 \( 1 + (-1.55e3 - 206. i)T \)
good3 \( 1 + (13.9 + 24.1i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-34.6 - 59.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + 195.T + 1.68e4T^{2} \)
11 \( 1 + 137.T + 1.61e5T^{2} \)
13 \( 1 + (398. - 689. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (784. + 1.35e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (-208. + 361. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-155. + 270. i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + 1.03e4T + 2.86e7T^{2} \)
37 \( 1 - 3.32e3T + 6.93e7T^{2} \)
41 \( 1 + (-2.54e3 - 4.39e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (4.08e3 + 7.08e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.01e3 - 1.76e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (6.07e3 - 1.05e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.36e3 - 4.09e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (4.31e3 - 7.46e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (6.73e3 - 1.16e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-3.50e4 - 6.07e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (-9.04e3 - 1.56e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-4.40e4 - 7.62e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.15e5T + 3.93e9T^{2} \)
89 \( 1 + (1.29e4 - 2.24e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (6.50e4 + 1.12e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−16.07285377255727331384213183893, −14.22637811935392637148645222021, −13.44872149077798826194593006468, −12.49776511931535850433727901666, −11.29185923261483164501972274759, −9.573499614307823748974717426921, −7.14628014311337779732830965922, −6.81280542406333285081417133425, −5.65502959465808554366332866337, −2.59808515315668009418645366336, 0.16426761304176881503812719366, 3.46872945919473612392765571808, 5.04077866330552632402432359719, 5.91360057966664102129150415400, 9.204129352906288044298025136738, 9.862771216679687195400064177307, 10.83242781610871028913943741920, 12.44499753745067016799010062707, 13.12434781651442484523730952931, 15.05745364804765564191304500621

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