Properties

Degree $2$
Conductor $38$
Sign $-0.193 - 0.981i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.75 − 1.36i)2-s + (3.93 + 22.3i)3-s + (12.2 + 10.2i)4-s + (61.0 − 51.2i)5-s + (15.7 − 89.2i)6-s + (−95.8 + 166. i)7-s + (−32.0 − 55.4i)8-s + (−253. + 92.3i)9-s + (−299. + 109. i)10-s + (179. + 310. i)11-s + (−181. + 313. i)12-s + (−77.0 + 437. i)13-s + (587. − 493. i)14-s + (1.38e3 + 1.15e3i)15-s + (44.4 + 252. i)16-s + (−931. − 339. i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.252 + 1.43i)3-s + (0.383 + 0.321i)4-s + (1.09 − 0.916i)5-s + (0.178 − 1.01i)6-s + (−0.739 + 1.28i)7-s + (−0.176 − 0.306i)8-s + (−1.04 + 0.379i)9-s + (−0.947 + 0.344i)10-s + (0.446 + 0.773i)11-s + (−0.363 + 0.629i)12-s + (−0.126 + 0.717i)13-s + (0.801 − 0.672i)14-s + (1.58 + 1.33i)15-s + (0.0434 + 0.246i)16-s + (−0.782 − 0.284i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.193 - 0.981i$
Motivic weight: \(5\)
Character: $\chi_{38} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.193 - 0.981i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.809751 + 0.984818i\)
\(L(\frac12)\) \(\approx\) \(0.809751 + 0.984818i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.75 + 1.36i)T \)
19 \( 1 + (1.31e3 - 865. i)T \)
good3 \( 1 + (-3.93 - 22.3i)T + (-228. + 83.1i)T^{2} \)
5 \( 1 + (-61.0 + 51.2i)T + (542. - 3.07e3i)T^{2} \)
7 \( 1 + (95.8 - 166. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-179. - 310. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (77.0 - 437. i)T + (-3.48e5 - 1.26e5i)T^{2} \)
17 \( 1 + (931. + 339. i)T + (1.08e6 + 9.12e5i)T^{2} \)
23 \( 1 + (-1.02e3 - 860. i)T + (1.11e6 + 6.33e6i)T^{2} \)
29 \( 1 + (-4.76e3 + 1.73e3i)T + (1.57e7 - 1.31e7i)T^{2} \)
31 \( 1 + (-3.27e3 + 5.67e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 1.32e4T + 6.93e7T^{2} \)
41 \( 1 + (1.71e3 + 9.73e3i)T + (-1.08e8 + 3.96e7i)T^{2} \)
43 \( 1 + (-8.63e3 + 7.24e3i)T + (2.55e7 - 1.44e8i)T^{2} \)
47 \( 1 + (2.64e4 - 9.63e3i)T + (1.75e8 - 1.47e8i)T^{2} \)
53 \( 1 + (-2.34e4 - 1.96e4i)T + (7.26e7 + 4.11e8i)T^{2} \)
59 \( 1 + (-3.86e4 - 1.40e4i)T + (5.47e8 + 4.59e8i)T^{2} \)
61 \( 1 + (1.22e4 + 1.03e4i)T + (1.46e8 + 8.31e8i)T^{2} \)
67 \( 1 + (2.08e4 - 7.58e3i)T + (1.03e9 - 8.67e8i)T^{2} \)
71 \( 1 + (-1.27e4 + 1.06e4i)T + (3.13e8 - 1.77e9i)T^{2} \)
73 \( 1 + (-1.05e3 - 5.99e3i)T + (-1.94e9 + 7.09e8i)T^{2} \)
79 \( 1 + (1.57e4 + 8.91e4i)T + (-2.89e9 + 1.05e9i)T^{2} \)
83 \( 1 + (-5.04e4 + 8.74e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-1.90e4 + 1.08e5i)T + (-5.24e9 - 1.90e9i)T^{2} \)
97 \( 1 + (-7.68e4 - 2.79e4i)T + (6.57e9 + 5.51e9i)T^{2} \)
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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

−15.81196485887924461829078311839, −14.82399017856863457269995485553, −13.09861668224859004865362470390, −11.86060677223005348180810799733, −10.11450406191731310494818729719, −9.326550032571696690196975615674, −8.869741744795874366078960275228, −6.12228321988978055630866135651, −4.50973574501255352369787023659, −2.30746764115570162124907178238, 0.899697631745169269183301659300, 2.70041872433812858539677759927, 6.51462290298638832705527202682, 6.72835164309404031987688224390, 8.317309188933994722623358717142, 9.986902914081763272786270746507, 10.98688148519810062901526311954, 12.98215311480584290914374189021, 13.64287419401907015723389046090, 14.65723505712599669481724664208

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