Properties

Label 2-38-19.5-c9-0-10
Degree $2$
Conductor $38$
Sign $-0.469 + 0.882i$
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 + (7.21 − 40.8i)3-s + (196. − 164. i)4-s + (488. + 410. i)5-s + (115. + 654. i)6-s + (−4.41e3 − 7.64e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (1.68e4 + 6.14e3i)9-s + (−9.59e3 − 3.49e3i)10-s + (5.46e3 − 9.46e3i)11-s + (−5.31e3 − 9.20e3i)12-s + (1.70e4 + 9.69e4i)13-s + (1.08e5 + 9.07e4i)14-s + (2.03e4 − 1.70e4i)15-s + (1.13e4 − 6.45e4i)16-s + (−3.41e5 + 1.24e5i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.0513 − 0.291i)3-s + (0.383 − 0.321i)4-s + (0.349 + 0.293i)5-s + (0.0363 + 0.206i)6-s + (−0.694 − 1.20i)7-s + (−0.176 + 0.306i)8-s + (0.857 + 0.312i)9-s + (−0.303 − 0.110i)10-s + (0.112 − 0.194i)11-s + (−0.0739 − 0.128i)12-s + (0.165 + 0.941i)13-s + (0.752 + 0.631i)14-s + (0.103 − 0.0868i)15-s + (0.0434 − 0.246i)16-s + (−0.991 + 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.469 + 0.882i$
Analytic conductor: \(19.5713\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ -0.469 + 0.882i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.433491 - 0.721927i\)
\(L(\frac12)\) \(\approx\) \(0.433491 - 0.721927i\)
\(L(\frac{11}{2})\) 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 \( 1 + (3.63e5 + 4.36e5i)T \)
good3 \( 1 + (-7.21 + 40.8i)T + (-1.84e4 - 6.73e3i)T^{2} \)
5 \( 1 + (-488. - 410. i)T + (3.39e5 + 1.92e6i)T^{2} \)
7 \( 1 + (4.41e3 + 7.64e3i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 + (-5.46e3 + 9.46e3i)T + (-1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 + (-1.70e4 - 9.69e4i)T + (-9.96e9 + 3.62e9i)T^{2} \)
17 \( 1 + (3.41e5 - 1.24e5i)T + (9.08e10 - 7.62e10i)T^{2} \)
23 \( 1 + (-1.57e6 + 1.32e6i)T + (3.12e11 - 1.77e12i)T^{2} \)
29 \( 1 + (4.04e6 + 1.47e6i)T + (1.11e13 + 9.32e12i)T^{2} \)
31 \( 1 + (3.78e6 + 6.55e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + 1.61e7T + 1.29e14T^{2} \)
41 \( 1 + (3.41e5 - 1.93e6i)T + (-3.07e14 - 1.11e14i)T^{2} \)
43 \( 1 + (1.56e7 + 1.31e7i)T + (8.72e13 + 4.94e14i)T^{2} \)
47 \( 1 + (-1.91e7 - 6.98e6i)T + (8.57e14 + 7.19e14i)T^{2} \)
53 \( 1 + (-6.50e7 + 5.45e7i)T + (5.72e14 - 3.24e15i)T^{2} \)
59 \( 1 + (1.32e8 - 4.82e7i)T + (6.63e15 - 5.56e15i)T^{2} \)
61 \( 1 + (2.51e7 - 2.10e7i)T + (2.03e15 - 1.15e16i)T^{2} \)
67 \( 1 + (-2.50e8 - 9.10e7i)T + (2.08e16 + 1.74e16i)T^{2} \)
71 \( 1 + (1.04e8 + 8.79e7i)T + (7.96e15 + 4.51e16i)T^{2} \)
73 \( 1 + (-3.79e7 + 2.15e8i)T + (-5.53e16 - 2.01e16i)T^{2} \)
79 \( 1 + (1.05e8 - 5.97e8i)T + (-1.12e17 - 4.09e16i)T^{2} \)
83 \( 1 + (3.67e8 + 6.37e8i)T + (-9.34e16 + 1.61e17i)T^{2} \)
89 \( 1 + (6.90e7 + 3.91e8i)T + (-3.29e17 + 1.19e17i)T^{2} \)
97 \( 1 + (1.72e8 - 6.28e7i)T + (5.82e17 - 4.88e17i)T^{2} \)
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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

−13.78097718920450462031604228512, −12.97346846942699572334079439054, −11.05771759443461410835335283763, −10.16615424929085761580800529046, −8.890623196359204803245902818028, −7.16363743887133745561333930714, −6.54971420449640092980848787510, −4.20024226365120041392970492968, −2.02831016023750362387565750505, −0.37068856433675987005144385263, 1.64424900014212803827533938838, 3.33318126599922309603806159614, 5.44434544454810900505869591149, 7.01115644369758937165745047496, 8.833929453645799217283337956113, 9.517997109918026524843182121959, 10.79176011887518628116256578679, 12.37127531432485586707383862830, 13.08620891647703304104495319533, 15.18787529568705169914966405990

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