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Properties

Label 2-195-195.194-c0-0-2
Degree $2$
Conductor $195$
Sign $0.707 - 0.707i$
Analytic cond. $0.0973176$
Root an. cond. $0.311957$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − i·3-s − 1.00·4-s + (0.707 − 0.707i)5-s + 1.41·6-s − 9-s + (1.00 + 1.00i)10-s − 1.41·11-s + 1.00i·12-s + i·13-s + (−0.707 − 0.707i)15-s − 0.999·16-s − 1.41i·18-s + (−0.707 + 0.707i)20-s − 2.00i·22-s + ⋯
L(s)  = 1  + 1.41i·2-s − i·3-s − 1.00·4-s + (0.707 − 0.707i)5-s + 1.41·6-s − 9-s + (1.00 + 1.00i)10-s − 1.41·11-s + 1.00i·12-s + i·13-s + (−0.707 − 0.707i)15-s − 0.999·16-s − 1.41i·18-s + (−0.707 + 0.707i)20-s − 2.00i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.0973176\)
Root analytic conductor: \(0.311957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7046978936\)
\(L(\frac12)\) \(\approx\) \(0.7046978936\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 - iT \)
good2 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−13.21504086688074319830768585323, −12.25469729002513209592688854700, −10.99106231624300302462606624737, −9.440866714430305903006383467153, −8.469663218549072951032124486086, −7.70808450273608444746121703780, −6.67272123123419501087238197541, −5.76768944560953259787734307338, −4.88969764023033145209072299567, −2.23288252098447639708228577616, 2.52383809435635533548851648622, 3.29967813781756604352129154086, 4.84633011308720520843676092457, 6.00811072252381319557068017303, 7.80245856384292925587611057570, 9.257280885009195301406715077245, 10.05477952043037237846160495428, 10.67969344315519172032264834362, 11.18125484904754212596057441347, 12.55625969666550831010351960632

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