Properties

Label 2-38-19.4-c9-0-6
Degree $2$
Conductor $38$
Sign $0.864 - 0.503i$
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  + (−15.0 − 5.47i)2-s + (19.6 + 111. i)3-s + (196. + 164. i)4-s + (−757. + 635. i)5-s + (313. − 1.77e3i)6-s + (3.78e3 − 6.55e3i)7-s + (−2.04e3 − 3.54e3i)8-s + (6.51e3 − 2.37e3i)9-s + (1.48e4 − 5.40e3i)10-s + (−7.00e3 − 1.21e4i)11-s + (−1.44e4 + 2.50e4i)12-s + (−1.07e3 + 6.07e3i)13-s + (−9.28e4 + 7.78e4i)14-s + (−8.54e4 − 7.17e4i)15-s + (1.13e4 + 6.45e4i)16-s + (1.96e5 + 7.14e4i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.139 + 0.792i)3-s + (0.383 + 0.321i)4-s + (−0.541 + 0.454i)5-s + (0.0988 − 0.560i)6-s + (0.596 − 1.03i)7-s + (−0.176 − 0.306i)8-s + (0.330 − 0.120i)9-s + (0.469 − 0.171i)10-s + (−0.144 − 0.249i)11-s + (−0.201 + 0.348i)12-s + (−0.0104 + 0.0589i)13-s + (−0.645 + 0.541i)14-s + (−0.435 − 0.365i)15-s + (0.0434 + 0.246i)16-s + (0.570 + 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.864 - 0.503i$
Analytic conductor: \(19.5713\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ 0.864 - 0.503i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.39908 + 0.377861i\)
\(L(\frac12)\) \(\approx\) \(1.39908 + 0.377861i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (15.0 + 5.47i)T \)
19 \( 1 + (4.50e5 + 3.46e5i)T \)
good3 \( 1 + (-19.6 - 111. i)T + (-1.84e4 + 6.73e3i)T^{2} \)
5 \( 1 + (757. - 635. i)T + (3.39e5 - 1.92e6i)T^{2} \)
7 \( 1 + (-3.78e3 + 6.55e3i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (7.00e3 + 1.21e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + (1.07e3 - 6.07e3i)T + (-9.96e9 - 3.62e9i)T^{2} \)
17 \( 1 + (-1.96e5 - 7.14e4i)T + (9.08e10 + 7.62e10i)T^{2} \)
23 \( 1 + (-1.87e6 - 1.57e6i)T + (3.12e11 + 1.77e12i)T^{2} \)
29 \( 1 + (-6.41e6 + 2.33e6i)T + (1.11e13 - 9.32e12i)T^{2} \)
31 \( 1 + (-6.31e5 + 1.09e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 - 8.20e6T + 1.29e14T^{2} \)
41 \( 1 + (-2.75e6 - 1.56e7i)T + (-3.07e14 + 1.11e14i)T^{2} \)
43 \( 1 + (1.56e7 - 1.31e7i)T + (8.72e13 - 4.94e14i)T^{2} \)
47 \( 1 + (7.72e6 - 2.81e6i)T + (8.57e14 - 7.19e14i)T^{2} \)
53 \( 1 + (3.39e7 + 2.84e7i)T + (5.72e14 + 3.24e15i)T^{2} \)
59 \( 1 + (-1.13e8 - 4.13e7i)T + (6.63e15 + 5.56e15i)T^{2} \)
61 \( 1 + (6.22e6 + 5.22e6i)T + (2.03e15 + 1.15e16i)T^{2} \)
67 \( 1 + (2.21e7 - 8.04e6i)T + (2.08e16 - 1.74e16i)T^{2} \)
71 \( 1 + (-1.85e8 + 1.55e8i)T + (7.96e15 - 4.51e16i)T^{2} \)
73 \( 1 + (-2.14e7 - 1.21e8i)T + (-5.53e16 + 2.01e16i)T^{2} \)
79 \( 1 + (-7.39e7 - 4.19e8i)T + (-1.12e17 + 4.09e16i)T^{2} \)
83 \( 1 + (-3.05e8 + 5.28e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (-5.90e7 + 3.34e8i)T + (-3.29e17 - 1.19e17i)T^{2} \)
97 \( 1 + (-7.12e8 - 2.59e8i)T + (5.82e17 + 4.88e17i)T^{2} \)
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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

−14.72403842957338477350011349938, −13.19387253632172298976393274399, −11.44057741032439204594173368428, −10.66347005517003061355096864440, −9.609312508756403751482731819084, −8.094570714101960124878012202071, −6.95275257069313351172130507377, −4.54442364209274847380099680361, −3.27426669386457021900808139140, −1.02902327035092483854295917091, 0.898410787097594883655526086191, 2.32378128550737818382846362193, 4.89413260420488245587089707728, 6.63161958682579689650539753592, 7.997622303961022581983384578019, 8.711621813909607665420017527617, 10.42314809121382138922532641331, 11.99146857081395892039535480978, 12.67529079305632981553041980902, 14.39482893393128319311148580934

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