Properties

Degree $2$
Conductor $38$
Sign $-0.840 + 0.541i$
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 − 15.3i·3-s − 8.00·4-s + 41.6·5-s − 43.4·6-s − 62.4·7-s + 22.6i·8-s − 154.·9-s − 117. i·10-s + 122.·11-s + 122. i·12-s + 68.9i·13-s + 176. i·14-s − 638. i·15-s + 64.0·16-s + 297.·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.70i·3-s − 0.500·4-s + 1.66·5-s − 1.20·6-s − 1.27·7-s + 0.353i·8-s − 1.90·9-s − 1.17i·10-s + 1.01·11-s + 0.852i·12-s + 0.408i·13-s + 0.900i·14-s − 2.83i·15-s + 0.250·16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.840 + 0.541i$
Motivic weight: \(4\)
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ -0.840 + 0.541i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.436367 - 1.48188i\)
\(L(\frac12)\) \(\approx\) \(0.436367 - 1.48188i\)
\(L(3)\) 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 \( 1 + (195. + 303. i)T \)
good3 \( 1 + 15.3iT - 81T^{2} \)
5 \( 1 - 41.6T + 625T^{2} \)
7 \( 1 + 62.4T + 2.40e3T^{2} \)
11 \( 1 - 122.T + 1.46e4T^{2} \)
13 \( 1 - 68.9iT - 2.85e4T^{2} \)
17 \( 1 - 297.T + 8.35e4T^{2} \)
23 \( 1 - 268.T + 2.79e5T^{2} \)
29 \( 1 + 561. iT - 7.07e5T^{2} \)
31 \( 1 + 252. iT - 9.23e5T^{2} \)
37 \( 1 - 2.40e3iT - 1.87e6T^{2} \)
41 \( 1 + 690. iT - 2.82e6T^{2} \)
43 \( 1 - 218.T + 3.41e6T^{2} \)
47 \( 1 + 83.1T + 4.87e6T^{2} \)
53 \( 1 - 4.38e3iT - 7.89e6T^{2} \)
59 \( 1 + 476. iT - 1.21e7T^{2} \)
61 \( 1 - 3.96e3T + 1.38e7T^{2} \)
67 \( 1 - 5.11e3iT - 2.01e7T^{2} \)
71 \( 1 - 8.18e3iT - 2.54e7T^{2} \)
73 \( 1 + 7.34e3T + 2.83e7T^{2} \)
79 \( 1 + 1.48e3iT - 3.89e7T^{2} \)
83 \( 1 - 5.34e3T + 4.74e7T^{2} \)
89 \( 1 + 1.11e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.07e3iT - 8.85e7T^{2} \)
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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

−14.34570111565171575480150988690, −13.44015980082312426086863214634, −12.90124739540166186329210879024, −11.78727045842362205483605195228, −9.964953947314055989434160460092, −8.940874543636569378318745302429, −6.82087467934725025219984456751, −5.97416401507610809860735011924, −2.66338611845239073724008034363, −1.22290137124919347744828337817, 3.51547206206605362420800676381, 5.38000098369012329067096369423, 6.33767453472833263068159137589, 9.027302035996747475320046176994, 9.697564380039864465319740877324, 10.44619371237989371549386591668, 12.74665091902373497643016870598, 14.14984035605564647110187196846, 14.84719777626063257991396169105, 16.34057875873229856748284081765

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