Properties

Degree $2$
Conductor $38$
Sign $-0.162 - 0.986i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.06 + 2.57i)2-s + (−2.00 + 0.730i)3-s + (2.77 + 15.7i)4-s + (−1.87 + 10.6i)5-s + (−8.03 − 2.92i)6-s + (97.5 + 168. i)7-s + (−32.0 + 55.4i)8-s + (−182. + 153. i)9-s + (−33.0 + 27.7i)10-s + (8.54 − 14.8i)11-s + (−17.0 − 29.6i)12-s + (247. + 90.0i)13-s + (−135. + 768. i)14-s + (−4.00 − 22.7i)15-s + (−240. + 87.5i)16-s + (−442. − 371. i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−0.128 + 0.0468i)3-s + (0.0868 + 0.492i)4-s + (−0.0335 + 0.190i)5-s + (−0.0911 − 0.0331i)6-s + (0.752 + 1.30i)7-s + (−0.176 + 0.306i)8-s + (−0.751 + 0.630i)9-s + (−0.104 + 0.0877i)10-s + (0.0212 − 0.0368i)11-s + (−0.0342 − 0.0593i)12-s + (0.405 + 0.147i)13-s + (−0.184 + 1.04i)14-s + (−0.00459 − 0.0260i)15-s + (−0.234 + 0.0855i)16-s + (−0.371 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.162 - 0.986i$
Motivic weight: \(5\)
Character: $\chi_{38} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.162 - 0.986i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.26658 + 1.49265i\)
\(L(\frac12)\) \(\approx\) \(1.26658 + 1.49265i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.06 - 2.57i)T \)
19 \( 1 + (-1.49e3 + 499. i)T \)
good3 \( 1 + (2.00 - 0.730i)T + (186. - 156. i)T^{2} \)
5 \( 1 + (1.87 - 10.6i)T + (-2.93e3 - 1.06e3i)T^{2} \)
7 \( 1 + (-97.5 - 168. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-8.54 + 14.8i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-247. - 90.0i)T + (2.84e5 + 2.38e5i)T^{2} \)
17 \( 1 + (442. + 371. i)T + (2.46e5 + 1.39e6i)T^{2} \)
23 \( 1 + (98.1 + 556. i)T + (-6.04e6 + 2.20e6i)T^{2} \)
29 \( 1 + (-5.14e3 + 4.31e3i)T + (3.56e6 - 2.01e7i)T^{2} \)
31 \( 1 + (4.64e3 + 8.03e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 1.71e3T + 6.93e7T^{2} \)
41 \( 1 + (1.14e3 - 416. i)T + (8.87e7 - 7.44e7i)T^{2} \)
43 \( 1 + (2.68e3 - 1.52e4i)T + (-1.38e8 - 5.02e7i)T^{2} \)
47 \( 1 + (-1.14e3 + 959. i)T + (3.98e7 - 2.25e8i)T^{2} \)
53 \( 1 + (-4.45e3 - 2.52e4i)T + (-3.92e8 + 1.43e8i)T^{2} \)
59 \( 1 + (-2.27e3 - 1.90e3i)T + (1.24e8 + 7.04e8i)T^{2} \)
61 \( 1 + (-1.99e3 - 1.12e4i)T + (-7.93e8 + 2.88e8i)T^{2} \)
67 \( 1 + (3.21e3 - 2.69e3i)T + (2.34e8 - 1.32e9i)T^{2} \)
71 \( 1 + (-1.17e4 + 6.63e4i)T + (-1.69e9 - 6.17e8i)T^{2} \)
73 \( 1 + (-4.28e4 + 1.56e4i)T + (1.58e9 - 1.33e9i)T^{2} \)
79 \( 1 + (3.14e4 - 1.14e4i)T + (2.35e9 - 1.97e9i)T^{2} \)
83 \( 1 + (3.34e4 + 5.79e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-8.08e4 - 2.94e4i)T + (4.27e9 + 3.58e9i)T^{2} \)
97 \( 1 + (-1.00e5 - 8.40e4i)T + (1.49e9 + 8.45e9i)T^{2} \)
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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

−15.46364704575510015718495034911, −14.53142146090921273524640167306, −13.44294894317400315678537826707, −11.91642817579687860892611573660, −11.14587682728364979779171596694, −9.027832693379812078166200016290, −7.87049381231470280176653379056, −6.04232567757734149216829221284, −4.90077819402782811814680093766, −2.60849608885178913777433256433, 1.06857757292724509442689158003, 3.56083430039467175185941351543, 5.12716365839665796937077653806, 6.88777047669407930014085169223, 8.607510622341350842429268557913, 10.36767617441891137087570618382, 11.32008183682810788566066238683, 12.51854932311603656303931342784, 13.88894609541651307771063609792, 14.54313618001030377719313048330

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