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Properties

Label 2-195-3.2-c2-0-0
Degree $2$
Conductor $195$
Sign $0.766 + 0.642i$
Analytic cond. $5.31336$
Root an. cond. $2.30507$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.53i·2-s + (−1.92 + 2.29i)3-s − 8.47·4-s − 2.23i·5-s + (−8.11 − 6.81i)6-s − 3.35·7-s − 15.7i·8-s + (−1.56 − 8.86i)9-s + 7.89·10-s − 0.969i·11-s + (16.3 − 19.4i)12-s + 3.60·13-s − 11.8i·14-s + (5.13 + 4.31i)15-s + 21.8·16-s + 16.8i·17-s + ⋯
L(s)  = 1  + 1.76i·2-s + (−0.642 + 0.766i)3-s − 2.11·4-s − 0.447i·5-s + (−1.35 − 1.13i)6-s − 0.478·7-s − 1.97i·8-s + (−0.173 − 0.984i)9-s + 0.789·10-s − 0.0881i·11-s + (1.36 − 1.62i)12-s + 0.277·13-s − 0.845i·14-s + (0.342 + 0.287i)15-s + 1.36·16-s + 0.993i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(5.31336\)
Root analytic conductor: \(2.30507\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0635971 - 0.0231491i\)
\(L(\frac12)\) \(\approx\) \(0.0635971 - 0.0231491i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.92 - 2.29i)T \)
5 \( 1 + 2.23iT \)
13 \( 1 - 3.60T \)
good2 \( 1 - 3.53iT - 4T^{2} \)
7 \( 1 + 3.35T + 49T^{2} \)
11 \( 1 + 0.969iT - 121T^{2} \)
17 \( 1 - 16.8iT - 289T^{2} \)
19 \( 1 + 36.1T + 361T^{2} \)
23 \( 1 + 29.4iT - 529T^{2} \)
29 \( 1 + 0.896iT - 841T^{2} \)
31 \( 1 + 19.4T + 961T^{2} \)
37 \( 1 - 44.9T + 1.36e3T^{2} \)
41 \( 1 - 25.4iT - 1.68e3T^{2} \)
43 \( 1 + 61.9T + 1.84e3T^{2} \)
47 \( 1 + 9.61iT - 2.20e3T^{2} \)
53 \( 1 - 13.4iT - 2.80e3T^{2} \)
59 \( 1 + 113. iT - 3.48e3T^{2} \)
61 \( 1 + 32.6T + 3.72e3T^{2} \)
67 \( 1 + 102.T + 4.48e3T^{2} \)
71 \( 1 - 27.4iT - 5.04e3T^{2} \)
73 \( 1 + 31.7T + 5.32e3T^{2} \)
79 \( 1 + 51.8T + 6.24e3T^{2} \)
83 \( 1 - 130. iT - 6.88e3T^{2} \)
89 \( 1 - 93.4iT - 7.92e3T^{2} \)
97 \( 1 - 60.7T + 9.40e3T^{2} \)
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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

−13.18126403344104114258153661371, −12.55736317140806604752083427029, −10.97494389171212260332187650226, −9.921648260525516247126740898074, −8.860141823841523080331897333051, −8.189094476466851706597484765284, −6.51953604741659610004215902516, −6.13935034786815387759684389520, −4.86301214144498934000842010427, −3.99633238064343085722296448116, 0.04181048395276118618830014725, 1.78009522684099417277119079881, 3.03639886418261468716774256625, 4.50128415943958269901906843689, 5.97245619711905223845809208867, 7.26263001949518927472467338988, 8.701797696897969994422571472032, 9.845497700713359508255592463442, 10.74079960673822753699688478510, 11.42962463114053479805652602283

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