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Properties

Label 2-38-19.12-c4-0-6
Degree $2$
Conductor $38$
Sign $0.328 + 0.944i$
Analytic cond. $3.92805$
Root an. cond. $1.98193$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 − 1.41i)2-s + (3.70 − 2.13i)3-s + (3.99 − 6.92i)4-s + (−17.7 − 30.7i)5-s + (6.04 − 10.4i)6-s + 57.6·7-s − 22.6i·8-s + (−31.3 + 54.3i)9-s + (−87.0 − 50.2i)10-s + 144.·11-s − 34.2i·12-s + (−165. − 95.5i)13-s + (141. − 81.5i)14-s + (−131. − 75.9i)15-s + (−32.0 − 55.4i)16-s + (273. + 474. i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.411 − 0.237i)3-s + (0.249 − 0.433i)4-s + (−0.710 − 1.23i)5-s + (0.168 − 0.291i)6-s + 1.17·7-s − 0.353i·8-s + (−0.387 + 0.670i)9-s + (−0.870 − 0.502i)10-s + 1.19·11-s − 0.237i·12-s + (−0.979 − 0.565i)13-s + (0.720 − 0.415i)14-s + (−0.584 − 0.337i)15-s + (−0.125 − 0.216i)16-s + (0.947 + 1.64i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(3.92805\)
Root analytic conductor: \(1.98193\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.76546 - 1.25497i\)
\(L(\frac12)\) \(\approx\) \(1.76546 - 1.25497i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.44 + 1.41i)T \)
19 \( 1 + (208. - 294. i)T \)
good3 \( 1 + (-3.70 + 2.13i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (17.7 + 30.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 57.6T + 2.40e3T^{2} \)
11 \( 1 - 144.T + 1.46e4T^{2} \)
13 \( 1 + (165. + 95.5i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-273. - 474. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-97.8 + 169. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (16.6 + 9.62i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 933. iT - 9.23e5T^{2} \)
37 \( 1 + 1.97e3iT - 1.87e6T^{2} \)
41 \( 1 + (-1.40e3 + 814. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.10e3 + 1.90e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (281. - 487. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (922. + 532. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-1.31e3 + 758. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.67e3 - 4.63e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (4.36e3 + 2.52e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (1.23e3 - 711. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-2.82e3 - 4.88e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-532. + 307. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 2.74e3T + 4.74e7T^{2} \)
89 \( 1 + (-4.64e3 - 2.68e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (8.50e3 - 4.90e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−14.84154759254081542115685782227, −14.30634432168646351084710971743, −12.66326807102820409842510179489, −12.06505593384754033165712932623, −10.66198358319655589442251174405, −8.687286456596363064817670858559, −7.80005247838491675554654747209, −5.38072745417983287687553269308, −4.08505322674806807497671467438, −1.57852801958087679408787028686, 3.00415703172333605966638594415, 4.54647330676985360873963158767, 6.67911489301189218524089048580, 7.76653034623996765795538272588, 9.422983987626311582979956472008, 11.45886855580479690016987867163, 11.77153344209382606124287554547, 14.04809754445339153712203803657, 14.65688036462626301076667219945, 15.18310704714278151263865403919

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