Properties

Label 2-38-19.11-c1-0-1
Degree $2$
Conductor $38$
Sign $0.996 - 0.0841i$
Analytic cond. $0.303431$
Root an. cond. $0.550846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (1.32 − 2.29i)3-s + (−0.499 − 0.866i)4-s + (−1.82 + 3.15i)5-s + (1.32 + 2.29i)6-s − 1.64·7-s + 0.999·8-s + (−2 − 3.46i)9-s + (−1.82 − 3.15i)10-s + 0.645·11-s − 2.64·12-s + (−1 − 1.73i)13-s + (0.822 − 1.42i)14-s + (4.82 + 8.35i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.763 − 1.32i)3-s + (−0.249 − 0.433i)4-s + (−0.815 + 1.41i)5-s + (0.540 + 0.935i)6-s − 0.622·7-s + 0.353·8-s + (−0.666 − 1.15i)9-s + (−0.576 − 0.998i)10-s + 0.194·11-s − 0.763·12-s + (−0.277 − 0.480i)13-s + (0.219 − 0.380i)14-s + (1.24 + 2.15i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.996 - 0.0841i$
Analytic conductor: \(0.303431\)
Root analytic conductor: \(0.550846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :1/2),\ 0.996 - 0.0841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.690544 + 0.0290894i\)
\(L(\frac12)\) \(\approx\) \(0.690544 + 0.0290894i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-4.32 - 0.559i)T \)
good3 \( 1 + (-1.32 + 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.64T + 7T^{2} \)
11 \( 1 - 0.645T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.82 + 3.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.82 - 3.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.354T + 31T^{2} \)
37 \( 1 - 5.64T + 37T^{2} \)
41 \( 1 + (5.14 - 8.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.354 - 0.613i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.82 + 8.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.29 + 7.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.46 - 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.32 - 4.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.64 + 11.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.14 - 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.14 + 12.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−16.25860646957614612136050226923, −14.97190219014638765594847093175, −14.27769097581464132747583438843, −13.06929975917556897435703687483, −11.68127102739973871787790019888, −10.04358212902967516378340566452, −8.262432187883424117866718095399, −7.30170867762992245211796335260, −6.49219872856213178823662708925, −3.06943908801571298717436507810, 3.58445348312736865712644994143, 4.76394696203279592635746947386, 7.999607704626304097391748492245, 9.168284283031185434774753366717, 9.758188665870075328531043182577, 11.45864257037850880312151225599, 12.60815650771659578767799653540, 13.99264088704240955943022219590, 15.62543662459517427307003054503, 16.13550783652575540731107203033

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