Properties

Degree $2$
Conductor $38$
Sign $-0.999 + 0.0387i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s + 7.80i·3-s − 8.00·4-s − 33.0·5-s − 22.0·6-s + 16.0·7-s − 22.6i·8-s + 20.1·9-s − 93.6i·10-s − 215.·11-s − 62.4i·12-s + 281. i·13-s + 45.4i·14-s − 258. i·15-s + 64.0·16-s + 226.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.867i·3-s − 0.500·4-s − 1.32·5-s − 0.613·6-s + 0.328·7-s − 0.353i·8-s + 0.248·9-s − 0.936i·10-s − 1.78·11-s − 0.433i·12-s + 1.66i·13-s + 0.232i·14-s − 1.14i·15-s + 0.250·16-s + 0.784·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0387i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.999 + 0.0387i$
Motivic weight: \(4\)
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ -0.999 + 0.0387i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0155537 - 0.802003i\)
\(L(\frac12)\) \(\approx\) \(0.0155537 - 0.802003i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
19 \( 1 + (-13.9 - 360. i)T \)
good3 \( 1 - 7.80iT - 81T^{2} \)
5 \( 1 + 33.0T + 625T^{2} \)
7 \( 1 - 16.0T + 2.40e3T^{2} \)
11 \( 1 + 215.T + 1.46e4T^{2} \)
13 \( 1 - 281. iT - 2.85e4T^{2} \)
17 \( 1 - 226.T + 8.35e4T^{2} \)
23 \( 1 - 414.T + 2.79e5T^{2} \)
29 \( 1 + 606. iT - 7.07e5T^{2} \)
31 \( 1 - 478. iT - 9.23e5T^{2} \)
37 \( 1 - 104. iT - 1.87e6T^{2} \)
41 \( 1 + 1.89e3iT - 2.82e6T^{2} \)
43 \( 1 - 315.T + 3.41e6T^{2} \)
47 \( 1 + 474.T + 4.87e6T^{2} \)
53 \( 1 - 774. iT - 7.89e6T^{2} \)
59 \( 1 + 567. iT - 1.21e7T^{2} \)
61 \( 1 - 4.79e3T + 1.38e7T^{2} \)
67 \( 1 + 4.46e3iT - 2.01e7T^{2} \)
71 \( 1 - 7.63e3iT - 2.54e7T^{2} \)
73 \( 1 - 8.39e3T + 2.83e7T^{2} \)
79 \( 1 - 9.70e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.05e4T + 4.74e7T^{2} \)
89 \( 1 - 1.02e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.54e4iT - 8.85e7T^{2} \)
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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

−15.95466063782112133522198189894, −15.34603367814436935349406273368, −14.19320065165023623846737036370, −12.56538116149668367564998660141, −11.17891234321650548528556370268, −9.887664685539706176412332400175, −8.317171134728054257926125755522, −7.28164823474562927200422926353, −5.07252253047712024793266339879, −3.89393897250238198753432320968, 0.56051810683536469085704849293, 2.99676737039772214886529090026, 5.04864726349963879376606642930, 7.54111090346297570572808495039, 8.129153141515518737636400381076, 10.31979767975201805991687228674, 11.38555878630890379595118880875, 12.69364924558045831202186445849, 13.14636067384833494790865700666, 15.02903363063193344973352300937

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