Properties

Degree $2$
Conductor $38$
Sign $-0.942 - 0.333i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.347 + 1.96i)2-s + (−4.43 + 3.71i)3-s + (−3.75 + 1.36i)4-s + (−5.82 − 2.11i)5-s + (−8.86 − 7.43i)6-s + (−5.61 + 9.71i)7-s + (−4 − 6.92i)8-s + (1.12 − 6.35i)9-s + (2.15 − 12.2i)10-s + (22.5 + 39.0i)11-s + (11.5 − 20.0i)12-s + (17.3 + 14.5i)13-s + (−21.0 − 7.67i)14-s + (33.6 − 12.2i)15-s + (12.2 − 10.2i)16-s + (−5.38 − 30.5i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−0.852 + 0.715i)3-s + (−0.469 + 0.171i)4-s + (−0.520 − 0.189i)5-s + (−0.602 − 0.505i)6-s + (−0.302 + 0.524i)7-s + (−0.176 − 0.306i)8-s + (0.0414 − 0.235i)9-s + (0.0680 − 0.386i)10-s + (0.618 + 1.07i)11-s + (0.278 − 0.481i)12-s + (0.370 + 0.310i)13-s + (−0.402 − 0.146i)14-s + (0.579 − 0.211i)15-s + (0.191 − 0.160i)16-s + (−0.0768 − 0.436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.942 - 0.333i$
Motivic weight: \(3\)
Character: $\chi_{38} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ -0.942 - 0.333i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.128418 + 0.747284i\)
\(L(\frac12)\) \(\approx\) \(0.128418 + 0.747284i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.347 - 1.96i)T \)
19 \( 1 + (-13.8 - 81.6i)T \)
good3 \( 1 + (4.43 - 3.71i)T + (4.68 - 26.5i)T^{2} \)
5 \( 1 + (5.82 + 2.11i)T + (95.7 + 80.3i)T^{2} \)
7 \( 1 + (5.61 - 9.71i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-22.5 - 39.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-17.3 - 14.5i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (5.38 + 30.5i)T + (-4.61e3 + 1.68e3i)T^{2} \)
23 \( 1 + (-135. + 49.3i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (-4.20 + 23.8i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (151. - 262. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 121.T + 5.06e4T^{2} \)
41 \( 1 + (79.4 - 66.6i)T + (1.19e4 - 6.78e4i)T^{2} \)
43 \( 1 + (-417. - 151. i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-83.6 + 474. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 + (-414. + 150. i)T + (1.14e5 - 9.56e4i)T^{2} \)
59 \( 1 + (-42.1 - 238. i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (-499. + 181. i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (126. - 716. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (-211. - 77.1i)T + (2.74e5 + 2.30e5i)T^{2} \)
73 \( 1 + (269. - 225. i)T + (6.75e4 - 3.83e5i)T^{2} \)
79 \( 1 + (461. - 387. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (124. - 215. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (1.27e3 + 1.06e3i)T + (1.22e5 + 6.94e5i)T^{2} \)
97 \( 1 + (146. + 833. i)T + (-8.57e5 + 3.12e5i)T^{2} \)
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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

−16.22686978361732949366561555528, −15.50381388927933149983506011508, −14.32844220049579940488893489880, −12.60166026201837055253896685604, −11.62504035200185809967345794201, −10.11097372233997018123723392350, −8.787279735645327815321534686386, −7.03082056966835447908473353477, −5.50470822912783472397258660228, −4.20506019753539861250375713101, 0.72500703526505515992456136430, 3.63908570122628391781077396456, 5.81624153764807371359160333986, 7.24393443774299274976560114983, 9.077724418634424336549749395395, 10.94908387053702126457314641687, 11.47572447295808224325269654938, 12.77519443374358589994279002149, 13.67744048245828949539798333312, 15.25809844936001176373766930813

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