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Properties

Label 2-195-65.9-c1-0-8
Degree $2$
Conductor $195$
Sign $0.820 + 0.571i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
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  + (0.729 + 0.421i)2-s + (−0.866 − 0.5i)3-s + (−0.644 − 1.11i)4-s + (2.23 − 0.0545i)5-s + (−0.421 − 0.729i)6-s + (−0.347 + 0.200i)7-s − 2.77i·8-s + (0.499 + 0.866i)9-s + (1.65 + 0.901i)10-s + (2.45 − 4.24i)11-s + 1.28i·12-s + (3.55 − 0.572i)13-s − 0.338·14-s + (−1.96 − 1.07i)15-s + (−0.121 + 0.211i)16-s + (−6.13 + 3.54i)17-s + ⋯
L(s)  = 1  + (0.516 + 0.297i)2-s + (−0.499 − 0.288i)3-s + (−0.322 − 0.558i)4-s + (0.999 − 0.0244i)5-s + (−0.172 − 0.297i)6-s + (−0.131 + 0.0758i)7-s − 0.980i·8-s + (0.166 + 0.288i)9-s + (0.523 + 0.285i)10-s + (0.739 − 1.28i)11-s + 0.372i·12-s + (0.987 − 0.158i)13-s − 0.0903·14-s + (−0.506 − 0.276i)15-s + (−0.0304 + 0.0528i)16-s + (−1.48 + 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.820 + 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34052 - 0.420418i\)
\(L(\frac12)\) \(\approx\) \(1.34052 - 0.420418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-2.23 + 0.0545i)T \)
13 \( 1 + (-3.55 + 0.572i)T \)
good2 \( 1 + (-0.729 - 0.421i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (0.347 - 0.200i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.45 + 4.24i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (6.13 - 3.54i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.12 - 1.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.861 - 0.497i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.94 - 6.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.30T + 31T^{2} \)
37 \( 1 + (-7.62 - 4.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.65 - 4.60i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.74 + 1.00i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.62iT - 47T^{2} \)
53 \( 1 + 10.8iT - 53T^{2} \)
59 \( 1 + (1.52 + 2.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.55 - 6.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.32 - 3.07i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.16 + 5.48i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.01iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 10.5iT - 83T^{2} \)
89 \( 1 + (-0.262 + 0.454i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.15 + 4.71i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.95320902460171975011897416766, −11.27445385321983340565861039315, −10.63782486957573025674666312498, −9.391983664621722590431121450922, −8.588115003422879360146422348060, −6.61047793783629160608933714741, −6.13029494554316867059363763309, −5.25589661673319232793838556603, −3.73291457783893469009258577114, −1.41014389330816918483573479927, 2.24362061526476853416454942105, 3.99315463040159407230945960144, 4.91303774704045439811203179054, 6.18737841293284665986596422735, 7.27986093404821553401477191056, 9.036168852484936962354012679808, 9.478034863414042217343070374288, 10.91672362581860585148867601541, 11.62379914179027310370968128124, 12.77146307268190170943750979991

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