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Properties

Label 2-195-3.2-c2-0-19
Degree $2$
Conductor $195$
Sign $-0.246 + 0.969i$
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  − 2.90i·2-s + (2.90 + 0.739i)3-s − 4.43·4-s + 2.23i·5-s + (2.14 − 8.44i)6-s + 2.78·7-s + 1.27i·8-s + (7.90 + 4.30i)9-s + 6.49·10-s − 19.6i·11-s + (−12.9 − 3.28i)12-s + 3.60·13-s − 8.09i·14-s + (−1.65 + 6.50i)15-s − 14.0·16-s − 13.3i·17-s + ⋯
L(s)  = 1  − 1.45i·2-s + (0.969 + 0.246i)3-s − 1.10·4-s + 0.447i·5-s + (0.358 − 1.40i)6-s + 0.398·7-s + 0.159i·8-s + (0.878 + 0.477i)9-s + 0.649·10-s − 1.78i·11-s + (−1.07 − 0.273i)12-s + 0.277·13-s − 0.578i·14-s + (−0.110 + 0.433i)15-s − 0.878·16-s − 0.785i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.246 + 0.969i$
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: \((2,\ 195,\ (\ :1),\ -0.246 + 0.969i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30993 - 1.68489i\)
\(L(\frac12)\) \(\approx\) \(1.30993 - 1.68489i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.90 - 0.739i)T \)
5 \( 1 - 2.23iT \)
13 \( 1 - 3.60T \)
good2 \( 1 + 2.90iT - 4T^{2} \)
7 \( 1 - 2.78T + 49T^{2} \)
11 \( 1 + 19.6iT - 121T^{2} \)
17 \( 1 + 13.3iT - 289T^{2} \)
19 \( 1 - 25.4T + 361T^{2} \)
23 \( 1 - 19.2iT - 529T^{2} \)
29 \( 1 - 46.4iT - 841T^{2} \)
31 \( 1 + 0.833T + 961T^{2} \)
37 \( 1 + 73.3T + 1.36e3T^{2} \)
41 \( 1 - 62.2iT - 1.68e3T^{2} \)
43 \( 1 + 8.66T + 1.84e3T^{2} \)
47 \( 1 + 4.59iT - 2.20e3T^{2} \)
53 \( 1 + 18.1iT - 2.80e3T^{2} \)
59 \( 1 - 7.60iT - 3.48e3T^{2} \)
61 \( 1 - 37.2T + 3.72e3T^{2} \)
67 \( 1 - 45.2T + 4.48e3T^{2} \)
71 \( 1 - 33.9iT - 5.04e3T^{2} \)
73 \( 1 - 72.0T + 5.32e3T^{2} \)
79 \( 1 + 150.T + 6.24e3T^{2} \)
83 \( 1 + 28.6iT - 6.88e3T^{2} \)
89 \( 1 + 50.7iT - 7.92e3T^{2} \)
97 \( 1 - 18.9T + 9.40e3T^{2} \)
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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

−11.64935173520126864952117192985, −11.11279647910536234756566227554, −10.15460894536505271764966655944, −9.223508661453499941167044579736, −8.350779500351387958589491757352, −7.04479345279104886219849538891, −5.17080683362010547098192497002, −3.48333778507090087971081623933, −3.05830776661058386597435957205, −1.37901311518696268257609036775, 1.98301627374457117377464796539, 4.13360937723393104881679989181, 5.20566389472761661364499382155, 6.67060743680787131499658824877, 7.51068251400140156879888755243, 8.244620191320584489651632482874, 9.169362239264404895184779057770, 10.14469885518487591807377642356, 11.93752975256219241429310005591, 12.85046254354954980856191744685

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