Properties

Label 2-195-13.12-c1-0-6
Degree $2$
Conductor $195$
Sign $0.554 + 0.832i$
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  i·2-s + 3-s + 4-s i·5-s i·6-s + 2i·7-s − 3i·8-s + 9-s − 10-s + 12-s + (−3 + 2i)13-s + 2·14-s i·15-s − 16-s + 2·17-s i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s + 0.5·4-s − 0.447i·5-s − 0.408i·6-s + 0.755i·7-s − 1.06i·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s + (−0.832 + 0.554i)13-s + 0.534·14-s − 0.258i·15-s − 0.250·16-s + 0.485·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38533 - 0.741408i\)
\(L(\frac12)\) \(\approx\) \(1.38533 - 0.741408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + iT \)
13 \( 1 + (3 - 2i)T \)
good2 \( 1 + iT - 2T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 6iT - 67T^{2} \)
71 \( 1 + 8iT - 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−12.10092389946497967094518845263, −11.73357396073929429459945207173, −10.26500708952233229644507372900, −9.565906579272753576393468975562, −8.480145042370734818966805905501, −7.34216665971660701418224122869, −6.08686581816896391275181852712, −4.52668775556950447886645155713, −3.03245265967709586286349457604, −1.86683055293904426504490175266, 2.30089519745610317539094762195, 3.78394649756968781007067772104, 5.45547671157826123522163666875, 6.67437241435915524858130345297, 7.59174351985760416099526883224, 8.187579500563309076026877242675, 9.816589303913704214365748237135, 10.52223744335013142535052849493, 11.69474380175466232146780880744, 12.72732018640175976671471463221

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