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Properties

Label 2-195-195.17-c1-0-21
Degree $2$
Conductor $195$
Sign $0.140 + 0.990i$
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  + (1.65 − 0.442i)2-s + (−0.744 − 1.56i)3-s + (0.796 − 0.459i)4-s + (0.908 − 2.04i)5-s + (−1.92 − 2.25i)6-s + (0.109 + 0.0293i)7-s + (−1.30 + 1.30i)8-s + (−1.89 + 2.32i)9-s + (0.595 − 3.77i)10-s + (2.05 − 3.55i)11-s + (−1.31 − 0.902i)12-s + (3.24 + 1.57i)13-s + 0.193·14-s + (−3.87 + 0.100i)15-s + (−2.49 + 4.32i)16-s + (−5.04 − 1.35i)17-s + ⋯
L(s)  = 1  + (1.16 − 0.312i)2-s + (−0.429 − 0.902i)3-s + (0.398 − 0.229i)4-s + (0.406 − 0.913i)5-s + (−0.783 − 0.919i)6-s + (0.0413 + 0.0110i)7-s + (−0.461 + 0.461i)8-s + (−0.630 + 0.776i)9-s + (0.188 − 1.19i)10-s + (0.619 − 1.07i)11-s + (−0.378 − 0.260i)12-s + (0.900 + 0.435i)13-s + 0.0517·14-s + (−0.999 + 0.0259i)15-s + (−0.624 + 1.08i)16-s + (−1.22 − 0.327i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.140 + 0.990i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.140 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37146 - 1.19016i\)
\(L(\frac12)\) \(\approx\) \(1.37146 - 1.19016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.744 + 1.56i)T \)
5 \( 1 + (-0.908 + 2.04i)T \)
13 \( 1 + (-3.24 - 1.57i)T \)
good2 \( 1 + (-1.65 + 0.442i)T + (1.73 - i)T^{2} \)
7 \( 1 + (-0.109 - 0.0293i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.05 + 3.55i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (5.04 + 1.35i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.97 - 6.88i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.69 + 1.79i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.00 - 3.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.89iT - 31T^{2} \)
37 \( 1 + (1.09 + 4.10i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.73 - 3.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.88 + 1.30i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.185 + 0.185i)T + 47iT^{2} \)
53 \( 1 + (1.94 + 1.94i)T + 53iT^{2} \)
59 \( 1 + (0.619 - 0.357i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.04 - 3.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.764 + 2.85i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.25 + 7.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.00 + 3.00i)T + 73iT^{2} \)
79 \( 1 - 4.84iT - 79T^{2} \)
83 \( 1 + (-2.04 + 2.04i)T - 83iT^{2} \)
89 \( 1 + (-2.79 - 1.61i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.144 - 0.0386i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.44225613186985197860582320796, −11.66060073070620272404361897633, −10.91722780950406204893151021330, −8.993234654022252951401595161925, −8.409849019391239362116571326367, −6.65193938024845476920193204051, −5.76949783042043710395568916508, −4.88070717969131496248375585512, −3.39747572671991804981409653456, −1.53971659223445429276964980258, 3.01785561299260260069446014100, 4.18997422398252507858364205258, 5.17328767197859174566792570363, 6.29250367053733083922459570679, 7.00979723088526560287525352889, 9.107721931175692747754787325021, 9.780440878996605878686961491267, 11.05946177430550722335425430870, 11.60536081635396021651528758059, 13.07110934681256180564141775780

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