Properties

Label 2-38-19.17-c9-0-1
Degree $2$
Conductor $38$
Sign $-0.460 - 0.887i$
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  + (−2.77 + 15.7i)2-s + (−110. − 93.0i)3-s + (−240. − 87.5i)4-s + (−143. + 52.2i)5-s + (1.77e3 − 1.48e3i)6-s + (−1.90e3 − 3.29e3i)7-s + (2.04e3 − 3.54e3i)8-s + (218. + 1.23e3i)9-s + (−424. − 2.40e3i)10-s + (3.85e4 − 6.67e4i)11-s + (1.85e4 + 3.20e4i)12-s + (−8.89e4 + 7.46e4i)13-s + (5.71e4 − 2.07e4i)14-s + (2.07e4 + 7.56e3i)15-s + (5.02e4 + 4.21e4i)16-s + (−1.07e5 + 6.09e5i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−0.790 − 0.662i)3-s + (−0.469 − 0.171i)4-s + (−0.102 + 0.0373i)5-s + (0.558 − 0.468i)6-s + (−0.299 − 0.518i)7-s + (0.176 − 0.306i)8-s + (0.0110 + 0.0628i)9-s + (−0.0134 − 0.0761i)10-s + (0.793 − 1.37i)11-s + (0.257 + 0.446i)12-s + (−0.863 + 0.724i)13-s + (0.397 − 0.144i)14-s + (0.105 + 0.0385i)15-s + (0.191 + 0.160i)16-s + (−0.312 + 1.76i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.460 - 0.887i$
Analytic conductor: \(19.5713\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ -0.460 - 0.887i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.247534 + 0.407349i\)
\(L(\frac12)\) \(\approx\) \(0.247534 + 0.407349i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.77 - 15.7i)T \)
19 \( 1 + (-5.67e5 - 1.65e4i)T \)
good3 \( 1 + (110. + 93.0i)T + (3.41e3 + 1.93e4i)T^{2} \)
5 \( 1 + (143. - 52.2i)T + (1.49e6 - 1.25e6i)T^{2} \)
7 \( 1 + (1.90e3 + 3.29e3i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 + (-3.85e4 + 6.67e4i)T + (-1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 + (8.89e4 - 7.46e4i)T + (1.84e9 - 1.04e10i)T^{2} \)
17 \( 1 + (1.07e5 - 6.09e5i)T + (-1.11e11 - 4.05e10i)T^{2} \)
23 \( 1 + (1.68e6 + 6.14e5i)T + (1.37e12 + 1.15e12i)T^{2} \)
29 \( 1 + (3.87e5 + 2.19e6i)T + (-1.36e13 + 4.96e12i)T^{2} \)
31 \( 1 + (-2.89e6 - 5.01e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 - 8.15e6T + 1.29e14T^{2} \)
41 \( 1 + (-1.61e7 - 1.35e7i)T + (5.68e13 + 3.22e14i)T^{2} \)
43 \( 1 + (1.50e7 - 5.46e6i)T + (3.85e14 - 3.23e14i)T^{2} \)
47 \( 1 + (-1.01e7 - 5.74e7i)T + (-1.05e15 + 3.82e14i)T^{2} \)
53 \( 1 + (-9.33e6 - 3.39e6i)T + (2.52e15 + 2.12e15i)T^{2} \)
59 \( 1 + (-2.28e7 + 1.29e8i)T + (-8.14e15 - 2.96e15i)T^{2} \)
61 \( 1 + (-1.76e6 - 6.42e5i)T + (8.95e15 + 7.51e15i)T^{2} \)
67 \( 1 + (-2.23e7 - 1.26e8i)T + (-2.55e16 + 9.30e15i)T^{2} \)
71 \( 1 + (-6.28e6 + 2.28e6i)T + (3.51e16 - 2.94e16i)T^{2} \)
73 \( 1 + (8.99e7 + 7.54e7i)T + (1.02e16 + 5.79e16i)T^{2} \)
79 \( 1 + (-1.33e8 - 1.11e8i)T + (2.08e16 + 1.18e17i)T^{2} \)
83 \( 1 + (-2.21e8 - 3.84e8i)T + (-9.34e16 + 1.61e17i)T^{2} \)
89 \( 1 + (8.52e8 - 7.15e8i)T + (6.08e16 - 3.45e17i)T^{2} \)
97 \( 1 + (-1.52e8 + 8.67e8i)T + (-7.14e17 - 2.60e17i)T^{2} \)
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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

−14.61480010373224907869120477238, −13.59841900364188957710638421478, −12.29004272381590360886265549683, −11.19758562325223234103474838196, −9.598488803721520017789928567935, −8.059966582366010568562803827076, −6.65571846103286026347851513614, −5.90742853535232833122545658329, −3.91830833124240934703576034375, −1.14777933652907327340393993728, 0.23296640162646057925962783369, 2.39603961891728324027043004485, 4.32711978816493138338113145280, 5.47380923463900632129937444715, 7.46415626213461527842612983384, 9.490693230341826422685750772649, 10.05575909471315000252834170528, 11.69549275661882378206737035529, 12.12112520020519283559934823960, 13.77763378714204577452564516139

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