Properties

Label 2-38-19.9-c9-0-5
Degree $2$
Conductor $38$
Sign $0.504 + 0.863i$
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 + (−69.9 + 58.6i)3-s + (−240. + 87.5i)4-s + (−954. − 347. i)5-s + (1.11e3 + 938. i)6-s + (−2.07e3 + 3.59e3i)7-s + (2.04e3 + 3.54e3i)8-s + (−1.97e3 + 1.11e4i)9-s + (−2.82e3 + 1.60e4i)10-s + (−3.03e4 − 5.25e4i)11-s + (1.16e4 − 2.02e4i)12-s + (5.50e4 + 4.61e4i)13-s + (6.23e4 + 2.27e4i)14-s + (8.71e4 − 3.17e4i)15-s + (5.02e4 − 4.21e4i)16-s + (3.63e4 + 2.05e5i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.498 + 0.418i)3-s + (−0.469 + 0.171i)4-s + (−0.683 − 0.248i)5-s + (0.352 + 0.295i)6-s + (−0.326 + 0.565i)7-s + (0.176 + 0.306i)8-s + (−0.100 + 0.568i)9-s + (−0.0892 + 0.506i)10-s + (−0.625 − 1.08i)11-s + (0.162 − 0.281i)12-s + (0.534 + 0.448i)13-s + (0.434 + 0.158i)14-s + (0.444 − 0.161i)15-s + (0.191 − 0.160i)16-s + (0.105 + 0.598i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.504 + 0.863i$
Analytic conductor: \(19.5713\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ 0.504 + 0.863i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.800792 - 0.459606i\)
\(L(\frac12)\) \(\approx\) \(0.800792 - 0.459606i\)
\(L(\frac{11}{2})\) 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 \( 1 + (-2.89e5 + 4.88e5i)T \)
good3 \( 1 + (69.9 - 58.6i)T + (3.41e3 - 1.93e4i)T^{2} \)
5 \( 1 + (954. + 347. i)T + (1.49e6 + 1.25e6i)T^{2} \)
7 \( 1 + (2.07e3 - 3.59e3i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (3.03e4 + 5.25e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + (-5.50e4 - 4.61e4i)T + (1.84e9 + 1.04e10i)T^{2} \)
17 \( 1 + (-3.63e4 - 2.05e5i)T + (-1.11e11 + 4.05e10i)T^{2} \)
23 \( 1 + (-1.88e6 + 6.86e5i)T + (1.37e12 - 1.15e12i)T^{2} \)
29 \( 1 + (-9.80e5 + 5.56e6i)T + (-1.36e13 - 4.96e12i)T^{2} \)
31 \( 1 + (1.46e6 - 2.53e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 - 1.21e6T + 1.29e14T^{2} \)
41 \( 1 + (-1.56e7 + 1.30e7i)T + (5.68e13 - 3.22e14i)T^{2} \)
43 \( 1 + (-3.43e7 - 1.24e7i)T + (3.85e14 + 3.23e14i)T^{2} \)
47 \( 1 + (8.73e6 - 4.95e7i)T + (-1.05e15 - 3.82e14i)T^{2} \)
53 \( 1 + (5.00e6 - 1.82e6i)T + (2.52e15 - 2.12e15i)T^{2} \)
59 \( 1 + (6.91e6 + 3.91e7i)T + (-8.14e15 + 2.96e15i)T^{2} \)
61 \( 1 + (2.27e7 - 8.29e6i)T + (8.95e15 - 7.51e15i)T^{2} \)
67 \( 1 + (1.60e7 - 9.08e7i)T + (-2.55e16 - 9.30e15i)T^{2} \)
71 \( 1 + (-1.92e8 - 6.99e7i)T + (3.51e16 + 2.94e16i)T^{2} \)
73 \( 1 + (4.23e6 - 3.55e6i)T + (1.02e16 - 5.79e16i)T^{2} \)
79 \( 1 + (-2.40e8 + 2.01e8i)T + (2.08e16 - 1.18e17i)T^{2} \)
83 \( 1 + (-2.08e8 + 3.61e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (-8.68e7 - 7.28e7i)T + (6.08e16 + 3.45e17i)T^{2} \)
97 \( 1 + (1.24e8 + 7.08e8i)T + (-7.14e17 + 2.60e17i)T^{2} \)
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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

−13.83594916373124318821550063501, −12.69050780107509307015985199727, −11.38949099323091076532302420600, −10.77781072654221528700103599000, −9.161953215540874045016244491706, −8.005985712472222741784197518192, −5.81312692783834814095799875018, −4.39739794197103549375632694796, −2.77591455307094105975955658065, −0.56235419906434658739854759049, 0.857316162544299742437974833410, 3.58117190730275228929828600364, 5.36281600164749660116598016620, 6.91099695657868774722381370035, 7.65067642851507533440467686139, 9.413878965196382277246776649418, 10.83025408247829332313013603214, 12.22473137818548821079254588269, 13.25938941955457402305275669312, 14.76150598140186490793511112580

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