Properties

Label 2-38-19.6-c9-0-6
Degree $2$
Conductor $38$
Sign $0.385 - 0.922i$
Analytic cond. $19.5713$
Root an. cond. $4.42395$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−12.2 + 10.2i)2-s + (221. + 80.5i)3-s + (44.4 − 252. i)4-s + (−138. − 786. i)5-s + (−3.54e3 + 1.28e3i)6-s + (−456. + 791. i)7-s + (2.04e3 + 3.54e3i)8-s + (2.74e4 + 2.30e4i)9-s + (9.78e3 + 8.20e3i)10-s + (2.33e4 + 4.05e4i)11-s + (3.01e4 − 5.22e4i)12-s + (1.95e4 − 7.12e3i)13-s + (−2.53e3 − 1.43e4i)14-s + (3.26e4 − 1.85e5i)15-s + (−6.15e4 − 2.24e4i)16-s + (2.15e5 − 1.80e5i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (1.57 + 0.574i)3-s + (0.0868 − 0.492i)4-s + (−0.0991 − 0.562i)5-s + (−1.11 + 0.406i)6-s + (−0.0719 + 0.124i)7-s + (0.176 + 0.306i)8-s + (1.39 + 1.17i)9-s + (0.309 + 0.259i)10-s + (0.481 + 0.834i)11-s + (0.419 − 0.727i)12-s + (0.190 − 0.0691i)13-s + (−0.0176 − 0.100i)14-s + (0.166 − 0.944i)15-s + (−0.234 − 0.0855i)16-s + (0.626 − 0.525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.385 - 0.922i$
Analytic conductor: \(19.5713\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ 0.385 - 0.922i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.14443 + 1.42875i\)
\(L(\frac12)\) \(\approx\) \(2.14443 + 1.42875i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (12.2 - 10.2i)T \)
19 \( 1 + (-4.82e5 - 2.99e5i)T \)
good3 \( 1 + (-221. - 80.5i)T + (1.50e4 + 1.26e4i)T^{2} \)
5 \( 1 + (138. + 786. i)T + (-1.83e6 + 6.68e5i)T^{2} \)
7 \( 1 + (456. - 791. i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (-2.33e4 - 4.05e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + (-1.95e4 + 7.12e3i)T + (8.12e9 - 6.81e9i)T^{2} \)
17 \( 1 + (-2.15e5 + 1.80e5i)T + (2.05e10 - 1.16e11i)T^{2} \)
23 \( 1 + (1.78e5 - 1.01e6i)T + (-1.69e12 - 6.16e11i)T^{2} \)
29 \( 1 + (-4.19e6 - 3.51e6i)T + (2.51e12 + 1.42e13i)T^{2} \)
31 \( 1 + (2.10e6 - 3.65e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + 6.71e6T + 1.29e14T^{2} \)
41 \( 1 + (1.19e7 + 4.36e6i)T + (2.50e14 + 2.10e14i)T^{2} \)
43 \( 1 + (1.12e6 + 6.35e6i)T + (-4.72e14 + 1.71e14i)T^{2} \)
47 \( 1 + (4.09e7 + 3.43e7i)T + (1.94e14 + 1.10e15i)T^{2} \)
53 \( 1 + (9.44e6 - 5.35e7i)T + (-3.10e15 - 1.12e15i)T^{2} \)
59 \( 1 + (-1.09e8 + 9.19e7i)T + (1.50e15 - 8.53e15i)T^{2} \)
61 \( 1 + (-4.76e6 + 2.70e7i)T + (-1.09e16 - 3.99e15i)T^{2} \)
67 \( 1 + (1.41e8 + 1.18e8i)T + (4.72e15 + 2.67e16i)T^{2} \)
71 \( 1 + (1.86e7 + 1.05e8i)T + (-4.30e16 + 1.56e16i)T^{2} \)
73 \( 1 + (-3.86e7 - 1.40e7i)T + (4.50e16 + 3.78e16i)T^{2} \)
79 \( 1 + (1.96e8 + 7.15e7i)T + (9.18e16 + 7.70e16i)T^{2} \)
83 \( 1 + (-1.51e8 + 2.61e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (-7.96e8 + 2.89e8i)T + (2.68e17 - 2.25e17i)T^{2} \)
97 \( 1 + (5.68e8 - 4.77e8i)T + (1.32e17 - 7.48e17i)T^{2} \)
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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

−14.61736072286830711452965905443, −13.81158261771083858007541938550, −12.24363644443703501450814831512, −10.17839073547048955085394942951, −9.284459549404786908755271093906, −8.404342563127675312893602555562, −7.22362387686280399093966345409, −4.94585973185222776427888783507, −3.32231988553534234614275247888, −1.52325817801020843012451553239, 1.10122623397469209957022171156, 2.65624994186056967078870977526, 3.63219456337516430606399630437, 6.72084771666197685037726504318, 7.963762821345656721220112544705, 8.867534362312178725444994168389, 10.11160691074103457788846438565, 11.62353959299588032150504296176, 13.06384067963036219267474837651, 14.02229007539567491958680201218

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