Properties

Label 2-38-19.9-c9-0-12
Degree $2$
Conductor $38$
Sign $-0.849 + 0.527i$
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 + (199. − 167. i)3-s + (−240. + 87.5i)4-s + (1.05e3 + 383. i)5-s + (−3.18e3 − 2.67e3i)6-s + (−1.72e3 + 2.98e3i)7-s + (2.04e3 + 3.54e3i)8-s + (8.34e3 − 4.73e4i)9-s + (3.11e3 − 1.76e4i)10-s + (−4.32e4 − 7.49e4i)11-s + (−3.33e4 + 5.76e4i)12-s + (8.21e3 + 6.89e3i)13-s + (5.18e4 + 1.88e4i)14-s + (2.74e5 − 9.98e4i)15-s + (5.02e4 − 4.21e4i)16-s + (−7.48e4 − 4.24e5i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (1.42 − 1.19i)3-s + (−0.469 + 0.171i)4-s + (0.754 + 0.274i)5-s + (−1.00 − 0.843i)6-s + (−0.271 + 0.470i)7-s + (0.176 + 0.306i)8-s + (0.423 − 2.40i)9-s + (0.0985 − 0.558i)10-s + (−0.890 − 1.54i)11-s + (−0.463 + 0.803i)12-s + (0.0797 + 0.0669i)13-s + (0.360 + 0.131i)14-s + (1.39 − 0.509i)15-s + (0.191 − 0.160i)16-s + (−0.217 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.849 + 0.527i$
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: \((2,\ 38,\ (\ :9/2),\ -0.849 + 0.527i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.755643 - 2.65046i\)
\(L(\frac12)\) \(\approx\) \(0.755643 - 2.65046i\)
\(L(\frac{11}{2})\) 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 \( 1 + (-4.96e5 - 2.76e5i)T \)
good3 \( 1 + (-199. + 167. i)T + (3.41e3 - 1.93e4i)T^{2} \)
5 \( 1 + (-1.05e3 - 383. i)T + (1.49e6 + 1.25e6i)T^{2} \)
7 \( 1 + (1.72e3 - 2.98e3i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (4.32e4 + 7.49e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + (-8.21e3 - 6.89e3i)T + (1.84e9 + 1.04e10i)T^{2} \)
17 \( 1 + (7.48e4 + 4.24e5i)T + (-1.11e11 + 4.05e10i)T^{2} \)
23 \( 1 + (1.03e5 - 3.76e4i)T + (1.37e12 - 1.15e12i)T^{2} \)
29 \( 1 + (6.24e5 - 3.54e6i)T + (-1.36e13 - 4.96e12i)T^{2} \)
31 \( 1 + (-2.39e6 + 4.15e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 - 1.01e7T + 1.29e14T^{2} \)
41 \( 1 + (1.31e7 - 1.10e7i)T + (5.68e13 - 3.22e14i)T^{2} \)
43 \( 1 + (-1.98e7 - 7.21e6i)T + (3.85e14 + 3.23e14i)T^{2} \)
47 \( 1 + (9.59e6 - 5.44e7i)T + (-1.05e15 - 3.82e14i)T^{2} \)
53 \( 1 + (-6.23e7 + 2.26e7i)T + (2.52e15 - 2.12e15i)T^{2} \)
59 \( 1 + (-7.14e6 - 4.05e7i)T + (-8.14e15 + 2.96e15i)T^{2} \)
61 \( 1 + (-1.69e8 + 6.18e7i)T + (8.95e15 - 7.51e15i)T^{2} \)
67 \( 1 + (2.29e7 - 1.29e8i)T + (-2.55e16 - 9.30e15i)T^{2} \)
71 \( 1 + (1.79e8 + 6.53e7i)T + (3.51e16 + 2.94e16i)T^{2} \)
73 \( 1 + (-1.02e8 + 8.59e7i)T + (1.02e16 - 5.79e16i)T^{2} \)
79 \( 1 + (1.88e8 - 1.58e8i)T + (2.08e16 - 1.18e17i)T^{2} \)
83 \( 1 + (-3.05e8 + 5.29e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (4.93e8 + 4.13e8i)T + (6.08e16 + 3.45e17i)T^{2} \)
97 \( 1 + (-2.00e7 - 1.13e8i)T + (-7.14e17 + 2.60e17i)T^{2} \)
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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

−13.65945817695517864391080496830, −12.94977020641724447933377497461, −11.56122964828339587836445080979, −9.766537260035778199196655846770, −8.769783423326503681347357568154, −7.65170885541647358161329512409, −5.97495116063781613479853690357, −3.12837068364002047080068150993, −2.44300749281307209061763040140, −0.899329629124987087387680551447, 2.18710595676609779047677811602, 3.97607072890720814312614944396, 5.20633357853104704086383226857, 7.37651774591738669162904354227, 8.595735037319800300716452111537, 9.801719393398946714835544350865, 10.24275012637073699597775273142, 13.07956134043154898567130175516, 13.80869130737622010559578542100, 15.05743644351283511766251010865

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