Properties

Degree $2$
Conductor $38$
Sign $0.770 - 0.637i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.347 + 1.96i)2-s + (2.93 − 2.45i)3-s + (−3.75 + 1.36i)4-s + (14.2 + 5.19i)5-s + (5.86 + 4.91i)6-s + (−0.806 + 1.39i)7-s + (−4 − 6.92i)8-s + (−2.14 + 12.1i)9-s + (−5.27 + 29.9i)10-s + (−19.2 − 33.3i)11-s + (−7.65 + 13.2i)12-s + (−3.06 − 2.56i)13-s + (−3.03 − 1.10i)14-s + (54.6 − 19.8i)15-s + (12.2 − 10.2i)16-s + (−15.5 − 88.0i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.563 − 0.473i)3-s + (−0.469 + 0.171i)4-s + (1.27 + 0.464i)5-s + (0.398 + 0.334i)6-s + (−0.0435 + 0.0754i)7-s + (−0.176 − 0.306i)8-s + (−0.0795 + 0.451i)9-s + (−0.166 + 0.946i)10-s + (−0.527 − 0.914i)11-s + (−0.184 + 0.318i)12-s + (−0.0652 − 0.0547i)13-s + (−0.0578 − 0.0210i)14-s + (0.940 − 0.342i)15-s + (0.191 − 0.160i)16-s + (−0.221 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.770 - 0.637i$
Motivic weight: \(3\)
Character: $\chi_{38} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.56896 + 0.564462i\)
\(L(\frac12)\) \(\approx\) \(1.56896 + 0.564462i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.347 - 1.96i)T \)
19 \( 1 + (77.1 + 30.1i)T \)
good3 \( 1 + (-2.93 + 2.45i)T + (4.68 - 26.5i)T^{2} \)
5 \( 1 + (-14.2 - 5.19i)T + (95.7 + 80.3i)T^{2} \)
7 \( 1 + (0.806 - 1.39i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (19.2 + 33.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (3.06 + 2.56i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (15.5 + 88.0i)T + (-4.61e3 + 1.68e3i)T^{2} \)
23 \( 1 + (63.4 - 23.0i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (46.2 - 262. i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (-95.4 + 165. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 137.T + 5.06e4T^{2} \)
41 \( 1 + (189. - 159. i)T + (1.19e4 - 6.78e4i)T^{2} \)
43 \( 1 + (-260. - 94.8i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-64.1 + 363. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 + (383. - 139. i)T + (1.14e5 - 9.56e4i)T^{2} \)
59 \( 1 + (102. + 579. i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (-757. + 275. i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (35.5 - 201. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (-671. - 244. i)T + (2.74e5 + 2.30e5i)T^{2} \)
73 \( 1 + (-81.6 + 68.4i)T + (6.75e4 - 3.83e5i)T^{2} \)
79 \( 1 + (393. - 329. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (439. - 760. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-362. - 303. i)T + (1.22e5 + 6.94e5i)T^{2} \)
97 \( 1 + (-285. - 1.61e3i)T + (-8.57e5 + 3.12e5i)T^{2} \)
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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

−15.94554242145738743206663908194, −14.44873342931215207558848159405, −13.74532105356787785188211799214, −13.00736211395284254361581468281, −10.90910999024407629509185567896, −9.451722288513373020794367609882, −8.149340336022872301092925350808, −6.71384417493337947999352898266, −5.37238790592604125968192353903, −2.58992815477760479930580117309, 2.15254292565823380774123943343, 4.26053320818227431816275109165, 6.02836937002210865859343573639, 8.494379169928872359903026325766, 9.684675796803956277691983210652, 10.37418007987861244346906486644, 12.31684416074886627621841965828, 13.23842473417217969016473502044, 14.37975933072605289138091937601, 15.40338982929702523858154153131

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