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Properties

Label 2-195-65.4-c1-0-2
Degree $2$
Conductor $195$
Sign $0.797 - 0.603i$
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  + (0.296 − 0.512i)2-s + (−0.866 − 0.5i)3-s + (0.824 + 1.42i)4-s + (−0.903 + 2.04i)5-s + (−0.512 + 0.296i)6-s + (0.828 + 1.43i)7-s + 2.16·8-s + (0.499 + 0.866i)9-s + (0.781 + 1.06i)10-s + (−1.22 − 0.706i)11-s − 1.64i·12-s + (0.0145 + 3.60i)13-s + 0.981·14-s + (1.80 − 1.31i)15-s + (−1.00 + 1.74i)16-s + (2.83 − 1.63i)17-s + ⋯
L(s)  = 1  + (0.209 − 0.362i)2-s + (−0.499 − 0.288i)3-s + (0.412 + 0.714i)4-s + (−0.404 + 0.914i)5-s + (−0.209 + 0.120i)6-s + (0.313 + 0.542i)7-s + 0.763·8-s + (0.166 + 0.288i)9-s + (0.247 + 0.337i)10-s + (−0.368 − 0.212i)11-s − 0.476i·12-s + (0.00404 + 0.999i)13-s + 0.262·14-s + (0.466 − 0.340i)15-s + (−0.252 + 0.437i)16-s + (0.688 − 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.797 - 0.603i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.797 - 0.603i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14273 + 0.383852i\)
\(L(\frac12)\) \(\approx\) \(1.14273 + 0.383852i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.903 - 2.04i)T \)
13 \( 1 + (-0.0145 - 3.60i)T \)
good2 \( 1 + (-0.296 + 0.512i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.828 - 1.43i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.22 + 0.706i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.83 + 1.63i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.29 + 4.21i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.70 + 3.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.252 + 0.438i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.791iT - 31T^{2} \)
37 \( 1 + (2.37 - 4.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.67 + 4.42i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.93 - 1.11i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.43T + 47T^{2} \)
53 \( 1 + 0.422iT - 53T^{2} \)
59 \( 1 + (-11.3 + 6.54i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.463 + 0.803i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.21 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.947 + 0.547i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + 9.25T + 79T^{2} \)
83 \( 1 - 4.02T + 83T^{2} \)
89 \( 1 + (0.517 + 0.299i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.57 + 2.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.10173964788180538110669356723, −11.80231653633297785568349778526, −11.04875043908584939033515247322, −9.933898774111487294204104837281, −8.348412153069201697647995942923, −7.38078719952077664402653503414, −6.58777947861440651721356251022, −5.07531862184227454539987939440, −3.54404831788231977761223664846, −2.30087746504972072311122588765, 1.21850054942205504329490792439, 3.84972283712217462816462086561, 5.24373736611301200728878252844, 5.69086228015661493394144942907, 7.36418127485420416864852439600, 8.069606824846718862409392424853, 9.794098182806564528478496385853, 10.31986072070940199733086378708, 11.50487427315609350497936938863, 12.27231253214591688496165889248

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