Properties

Degree $2$
Conductor $38$
Sign $0.984 - 0.175i$
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 + (19.8 + 16.6i)3-s + (−15.0 − 5.47i)4-s + (17.3 − 6.31i)5-s + (79.3 − 66.6i)6-s + (61.0 + 105. i)7-s + (−32 + 55.4i)8-s + (74.3 + 421. i)9-s + (−12.8 − 72.7i)10-s + (35.5 − 61.6i)11-s + (−207. − 358. i)12-s + (413. − 347. i)13-s + (458. − 167. i)14-s + (449. + 163. i)15-s + (196. + 164. i)16-s + (38.4 − 217. i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (1.27 + 1.06i)3-s + (−0.469 − 0.171i)4-s + (0.310 − 0.112i)5-s + (0.900 − 0.755i)6-s + (0.470 + 0.815i)7-s + (−0.176 + 0.306i)8-s + (0.305 + 1.73i)9-s + (−0.0405 − 0.229i)10-s + (0.0886 − 0.153i)11-s + (−0.415 − 0.719i)12-s + (0.679 − 0.569i)13-s + (0.625 − 0.227i)14-s + (0.515 + 0.187i)15-s + (0.191 + 0.160i)16-s + (0.0322 − 0.182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.984 - 0.175i$
Motivic weight: \(5\)
Character: $\chi_{38} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 0.984 - 0.175i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.48977 + 0.220377i\)
\(L(\frac12)\) \(\approx\) \(2.48977 + 0.220377i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-511. + 1.48e3i)T \)
good3 \( 1 + (-19.8 - 16.6i)T + (42.1 + 239. i)T^{2} \)
5 \( 1 + (-17.3 + 6.31i)T + (2.39e3 - 2.00e3i)T^{2} \)
7 \( 1 + (-61.0 - 105. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-35.5 + 61.6i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-413. + 347. i)T + (6.44e4 - 3.65e5i)T^{2} \)
17 \( 1 + (-38.4 + 217. i)T + (-1.33e6 - 4.85e5i)T^{2} \)
23 \( 1 + (3.86e3 + 1.40e3i)T + (4.93e6 + 4.13e6i)T^{2} \)
29 \( 1 + (-964. - 5.46e3i)T + (-1.92e7 + 7.01e6i)T^{2} \)
31 \( 1 + (3.77e3 + 6.53e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 6.11e3T + 6.93e7T^{2} \)
41 \( 1 + (1.56e4 + 1.30e4i)T + (2.01e7 + 1.14e8i)T^{2} \)
43 \( 1 + (4.79e3 - 1.74e3i)T + (1.12e8 - 9.44e7i)T^{2} \)
47 \( 1 + (-912. - 5.17e3i)T + (-2.15e8 + 7.84e7i)T^{2} \)
53 \( 1 + (-3.23e3 - 1.17e3i)T + (3.20e8 + 2.68e8i)T^{2} \)
59 \( 1 + (-9.08e3 + 5.15e4i)T + (-6.71e8 - 2.44e8i)T^{2} \)
61 \( 1 + (-4.15e4 - 1.51e4i)T + (6.46e8 + 5.42e8i)T^{2} \)
67 \( 1 + (-1.13e4 - 6.40e4i)T + (-1.26e9 + 4.61e8i)T^{2} \)
71 \( 1 + (3.70e4 - 1.34e4i)T + (1.38e9 - 1.15e9i)T^{2} \)
73 \( 1 + (5.82e3 + 4.88e3i)T + (3.59e8 + 2.04e9i)T^{2} \)
79 \( 1 + (-3.19e4 - 2.67e4i)T + (5.34e8 + 3.03e9i)T^{2} \)
83 \( 1 + (-1.78e4 - 3.09e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-6.82e4 + 5.72e4i)T + (9.69e8 - 5.49e9i)T^{2} \)
97 \( 1 + (2.90e3 - 1.64e4i)T + (-8.06e9 - 2.93e9i)T^{2} \)
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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

−15.13325275892449778723801451423, −14.18322356901204007592923354788, −13.16182125170411974017813634190, −11.50093322253077658605028732937, −10.17932464757210113758994821733, −9.129376552994174417192736271715, −8.255044744497218877786507522981, −5.32710765055701919181059014867, −3.73442439845071831422972529347, −2.28176336041672583499196088323, 1.65348924668444002075700189589, 3.83716146711718675635439896311, 6.30593812970964566921540048420, 7.58967571958328244899241686789, 8.392343140348219235029186223343, 9.898874393946851027383573688281, 12.01997587360054136885377178159, 13.54513909580916203283894510253, 13.89128208882876713057796083486, 14.85919514828050086203211985047

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