Properties

Degree $2$
Conductor $38$
Sign $-0.433 + 0.901i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.694 + 3.93i)2-s + (−21.8 + 18.3i)3-s + (−15.0 + 5.47i)4-s + (33.2 + 12.0i)5-s + (−87.3 − 73.2i)6-s + (−42.9 + 74.3i)7-s + (−32 − 55.4i)8-s + (98.8 − 560. i)9-s + (−24.5 + 139. i)10-s + (−299. − 519. i)11-s + (228. − 394. i)12-s + (417. + 350. i)13-s + (−322. − 117. i)14-s + (−946. + 344. i)15-s + (196. − 164. i)16-s + (252. + 1.43e3i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−1.40 + 1.17i)3-s + (−0.469 + 0.171i)4-s + (0.594 + 0.216i)5-s + (−0.990 − 0.831i)6-s + (−0.331 + 0.573i)7-s + (−0.176 − 0.306i)8-s + (0.406 − 2.30i)9-s + (−0.0776 + 0.440i)10-s + (−0.747 − 1.29i)11-s + (0.457 − 0.791i)12-s + (0.685 + 0.574i)13-s + (−0.440 − 0.160i)14-s + (−1.08 + 0.395i)15-s + (0.191 − 0.160i)16-s + (0.212 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.433 + 0.901i$
Motivic weight: \(5\)
Character: $\chi_{38} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.433 + 0.901i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.220815 - 0.351081i\)
\(L(\frac12)\) \(\approx\) \(0.220815 - 0.351081i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.694 - 3.93i)T \)
19 \( 1 + (1.57e3 - 94.1i)T \)
good3 \( 1 + (21.8 - 18.3i)T + (42.1 - 239. i)T^{2} \)
5 \( 1 + (-33.2 - 12.0i)T + (2.39e3 + 2.00e3i)T^{2} \)
7 \( 1 + (42.9 - 74.3i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (299. + 519. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-417. - 350. i)T + (6.44e4 + 3.65e5i)T^{2} \)
17 \( 1 + (-252. - 1.43e3i)T + (-1.33e6 + 4.85e5i)T^{2} \)
23 \( 1 + (3.44e3 - 1.25e3i)T + (4.93e6 - 4.13e6i)T^{2} \)
29 \( 1 + (-17.7 + 100. i)T + (-1.92e7 - 7.01e6i)T^{2} \)
31 \( 1 + (-440. + 763. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 8.65e3T + 6.93e7T^{2} \)
41 \( 1 + (-8.16e3 + 6.85e3i)T + (2.01e7 - 1.14e8i)T^{2} \)
43 \( 1 + (-4.71e3 - 1.71e3i)T + (1.12e8 + 9.44e7i)T^{2} \)
47 \( 1 + (-1.41e3 + 7.99e3i)T + (-2.15e8 - 7.84e7i)T^{2} \)
53 \( 1 + (2.43e4 - 8.85e3i)T + (3.20e8 - 2.68e8i)T^{2} \)
59 \( 1 + (-6.27e3 - 3.56e4i)T + (-6.71e8 + 2.44e8i)T^{2} \)
61 \( 1 + (3.69e4 - 1.34e4i)T + (6.46e8 - 5.42e8i)T^{2} \)
67 \( 1 + (2.37e3 - 1.34e4i)T + (-1.26e9 - 4.61e8i)T^{2} \)
71 \( 1 + (-7.02e4 - 2.55e4i)T + (1.38e9 + 1.15e9i)T^{2} \)
73 \( 1 + (4.52e4 - 3.79e4i)T + (3.59e8 - 2.04e9i)T^{2} \)
79 \( 1 + (-4.60e3 + 3.86e3i)T + (5.34e8 - 3.03e9i)T^{2} \)
83 \( 1 + (3.65e4 - 6.33e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-1.17e4 - 9.83e3i)T + (9.69e8 + 5.49e9i)T^{2} \)
97 \( 1 + (1.86e4 + 1.05e5i)T + (-8.06e9 + 2.93e9i)T^{2} \)
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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

−16.05647472830621698978720744265, −15.41896408312396122093067988367, −13.92204679374036238051026927137, −12.41142871865722518464186480428, −11.00946646762968178707412264596, −10.06901164994638638453787945418, −8.670546392957449832655225533829, −6.04053258925683381890321476513, −5.85372886937838220383939523656, −3.98122709717672794150443012095, 0.26294002770134863496618480389, 1.91130644582780361871557749589, 4.92413812886504463871848067500, 6.25314395270238521947660400216, 7.66129883674310502719481697168, 9.968847830273283850070351995798, 10.91694778546953193727221195180, 12.26180824624591569054431272149, 12.93919713368812620470458817642, 13.81930969910419765818092088549

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