Properties

Degree $2$
Conductor $38$
Sign $0.710 - 0.703i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.87 + 0.684i)2-s + (0.0408 − 0.231i)3-s + (3.06 − 2.57i)4-s + (8.45 + 7.09i)5-s + (0.0816 + 0.462i)6-s + (9.26 + 16.0i)7-s + (−4.00 + 6.92i)8-s + (25.3 + 9.21i)9-s + (−20.7 − 7.54i)10-s + (−6.98 + 12.1i)11-s + (−0.470 − 0.814i)12-s + (−9.37 − 53.1i)13-s + (−28.3 − 23.8i)14-s + (1.98 − 1.66i)15-s + (2.77 − 15.7i)16-s + (−7.65 + 2.78i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.00785 − 0.0445i)3-s + (0.383 − 0.321i)4-s + (0.756 + 0.634i)5-s + (0.00555 + 0.0314i)6-s + (0.500 + 0.866i)7-s + (−0.176 + 0.306i)8-s + (0.937 + 0.341i)9-s + (−0.655 − 0.238i)10-s + (−0.191 + 0.331i)11-s + (−0.0113 − 0.0195i)12-s + (−0.199 − 1.13i)13-s + (−0.542 − 0.454i)14-s + (0.0342 − 0.0287i)15-s + (0.0434 − 0.246i)16-s + (−0.109 + 0.0397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.710 - 0.703i$
Motivic weight: \(3\)
Character: $\chi_{38} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ 0.710 - 0.703i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.02580 + 0.421862i\)
\(L(\frac12)\) \(\approx\) \(1.02580 + 0.421862i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.87 - 0.684i)T \)
19 \( 1 + (31.6 + 76.5i)T \)
good3 \( 1 + (-0.0408 + 0.231i)T + (-25.3 - 9.23i)T^{2} \)
5 \( 1 + (-8.45 - 7.09i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (-9.26 - 16.0i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (6.98 - 12.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (9.37 + 53.1i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (7.65 - 2.78i)T + (3.76e3 - 3.15e3i)T^{2} \)
23 \( 1 + (93.7 - 78.6i)T + (2.11e3 - 1.19e4i)T^{2} \)
29 \( 1 + (-208. - 75.7i)T + (1.86e4 + 1.56e4i)T^{2} \)
31 \( 1 + (23.1 + 40.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 99.6T + 5.06e4T^{2} \)
41 \( 1 + (-55.1 + 312. i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (358. + 300. i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (-16.4 - 5.96i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 + (-33.9 + 28.4i)T + (2.58e4 - 1.46e5i)T^{2} \)
59 \( 1 + (599. - 218. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (-631. + 529. i)T + (3.94e4 - 2.23e5i)T^{2} \)
67 \( 1 + (-789. - 287. i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (442. + 371. i)T + (6.21e4 + 3.52e5i)T^{2} \)
73 \( 1 + (131. - 744. i)T + (-3.65e5 - 1.33e5i)T^{2} \)
79 \( 1 + (-58.9 + 334. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (-382. - 663. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-157. - 893. i)T + (-6.62e5 + 2.41e5i)T^{2} \)
97 \( 1 + (1.58e3 - 575. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−15.78591844381777380784103310637, −15.10381054687164238393519791256, −13.72774519170935331575969574199, −12.31123859666126691116235675569, −10.69488189432702480094641534663, −9.833202201489507063378245391013, −8.306971443600673582393832923864, −6.89784335785114823856780284815, −5.34594045531414258216042501909, −2.22922113351296215776659011489, 1.50395117912655908937421382552, 4.38973376250632290109302048616, 6.57315260910535363533522411651, 8.138300662964029956788989242216, 9.547880076678668880416241317877, 10.45764042802432146811244264312, 11.97564790995757878857012899284, 13.23982332173079155451596526445, 14.38135216172363238394719555374, 16.11092969355734133713706510312

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