Properties

Degree $2$
Conductor $38$
Sign $-0.711 + 0.702i$
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 − 10.8i·3-s − 8.00·4-s − 26.2·5-s + 30.6·6-s − 86.0·7-s − 22.6i·8-s − 36.2·9-s − 74.1i·10-s + 114.·11-s + 86.6i·12-s − 231. i·13-s − 243. i·14-s + 284. i·15-s + 64.0·16-s − 244.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.20i·3-s − 0.500·4-s − 1.04·5-s + 0.850·6-s − 1.75·7-s − 0.353i·8-s − 0.447·9-s − 0.741i·10-s + 0.948·11-s + 0.601i·12-s − 1.37i·13-s − 1.24i·14-s + 1.26i·15-s + 0.250·16-s − 0.845·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.711 + 0.702i$
Motivic weight: \(4\)
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ -0.711 + 0.702i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.179290 - 0.436726i\)
\(L(\frac12)\) \(\approx\) \(0.179290 - 0.436726i\)
\(L(3)\) 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 \( 1 + (-253. - 256. i)T \)
good3 \( 1 + 10.8iT - 81T^{2} \)
5 \( 1 + 26.2T + 625T^{2} \)
7 \( 1 + 86.0T + 2.40e3T^{2} \)
11 \( 1 - 114.T + 1.46e4T^{2} \)
13 \( 1 + 231. iT - 2.85e4T^{2} \)
17 \( 1 + 244.T + 8.35e4T^{2} \)
23 \( 1 + 269.T + 2.79e5T^{2} \)
29 \( 1 + 1.13e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.03e3iT - 9.23e5T^{2} \)
37 \( 1 - 302. iT - 1.87e6T^{2} \)
41 \( 1 + 1.76e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.31e3T + 3.41e6T^{2} \)
47 \( 1 + 835.T + 4.87e6T^{2} \)
53 \( 1 + 656. iT - 7.89e6T^{2} \)
59 \( 1 + 4.92e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.57e3T + 1.38e7T^{2} \)
67 \( 1 - 7.00e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.10e3iT - 2.54e7T^{2} \)
73 \( 1 - 7.01e3T + 2.83e7T^{2} \)
79 \( 1 - 1.26e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.02e3T + 4.74e7T^{2} \)
89 \( 1 + 3.74e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.30e4iT - 8.85e7T^{2} \)
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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

−15.39608429293913697667127134715, −13.73923621731755663904761526506, −12.77968769792361660589213327138, −11.97491907783936241633373337875, −9.875544502105352605620330510673, −8.244431150345076049044980754739, −7.11650355526117091921472838055, −6.17309335710984374026919559034, −3.58477140934765143233924250986, −0.31947384406043978224097145339, 3.45850268069678081648832984730, 4.34314650538667650447278291589, 6.75383165481682875017147314444, 9.097453835441313350267248816136, 9.678210252704333080779396324525, 11.13524612182684745054822787881, 12.10200708138997941326495708674, 13.51011563330840567110664098474, 15.04798910823789950301705333879, 16.07992495822755812302467286391

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