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Properties

Label 2-38-19.11-c3-0-0
Degree $2$
Conductor $38$
Sign $-0.980 - 0.194i$
Analytic cond. $2.24207$
Root an. cond. $1.49735$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−2.5 + 4.33i)3-s + (−1.99 − 3.46i)4-s + (−1.5 + 2.59i)5-s + (−5 − 8.66i)6-s − 32·7-s + 7.99·8-s + (0.999 + 1.73i)9-s + (−3 − 5.19i)10-s + 4·11-s + 20·12-s + (34.5 + 59.7i)13-s + (32 − 55.4i)14-s + (−7.50 − 12.9i)15-s + (−8 + 13.8i)16-s + (−9.5 + 16.4i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.481 + 0.833i)3-s + (−0.249 − 0.433i)4-s + (−0.134 + 0.232i)5-s + (−0.340 − 0.589i)6-s − 1.72·7-s + 0.353·8-s + (0.0370 + 0.0641i)9-s + (−0.0948 − 0.164i)10-s + 0.109·11-s + 0.481·12-s + (0.736 + 1.27i)13-s + (0.610 − 1.05i)14-s + (−0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.135 + 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.980 - 0.194i$
Analytic conductor: \(2.24207\)
Root analytic conductor: \(1.49735\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ -0.980 - 0.194i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0582265 + 0.592900i\)
\(L(\frac12)\) \(\approx\) \(0.0582265 + 0.592900i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
19 \( 1 + (-76 - 32.9i)T \)
good3 \( 1 + (2.5 - 4.33i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + 32T + 343T^{2} \)
11 \( 1 - 4T + 1.33e3T^{2} \)
13 \( 1 + (-34.5 - 59.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (9.5 - 16.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (33.5 + 58.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (25.5 + 44.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 132T + 2.97e4T^{2} \)
37 \( 1 + 14T + 5.06e4T^{2} \)
41 \( 1 + (-206.5 + 357. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (64.5 - 111. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-308.5 - 534. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (191.5 + 331. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-299.5 + 518. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-108.5 - 187. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-112.5 - 194. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (350.5 - 607. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (507.5 - 879. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (174.5 - 302. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 592T + 5.71e5T^{2} \)
89 \( 1 + (-674.5 - 1.16e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-306.5 + 530. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−16.10441543491617007622034977627, −15.84771353709858655258013640870, −14.23036720749007182959031374912, −12.91692362915614939792206810596, −11.20816321064741982072667981049, −9.981711083431891175352212229036, −9.118812386362574629286449912190, −7.06386732334685446277813405569, −5.85081849213172110714434792469, −3.91136547787382334409460444903, 0.60970772749884074980221549934, 3.32044710855048030118575657231, 6.00150474306587485868330129324, 7.35645399658817551427577787639, 9.107689955945650265477429993987, 10.30868495344333800147467302640, 11.83387670664792377248638250551, 12.79845573107374331924032903380, 13.39787648912094679968147737759, 15.60107345727530123403295476909

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