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Properties

Label 2-195-13.9-c1-0-9
Degree $2$
Conductor $195$
Sign $-0.755 + 0.655i$
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  + (1.36 − 2.36i)2-s + (−0.5 + 0.866i)3-s + (−2.73 − 4.73i)4-s + 5-s + (1.36 + 2.36i)6-s + (−0.866 − 1.5i)7-s − 9.46·8-s + (−0.499 − 0.866i)9-s + (1.36 − 2.36i)10-s + (1 − 1.73i)11-s + 5.46·12-s + (3.59 + 0.232i)13-s − 4.73·14-s + (−0.5 + 0.866i)15-s + (−7.46 + 12.9i)16-s + (1.36 + 2.36i)17-s + ⋯
L(s)  = 1  + (0.965 − 1.67i)2-s + (−0.288 + 0.499i)3-s + (−1.36 − 2.36i)4-s + 0.447·5-s + (0.557 + 0.965i)6-s + (−0.327 − 0.566i)7-s − 3.34·8-s + (−0.166 − 0.288i)9-s + (0.431 − 0.748i)10-s + (0.301 − 0.522i)11-s + 1.57·12-s + (0.997 + 0.0643i)13-s − 1.26·14-s + (−0.129 + 0.223i)15-s + (−1.86 + 3.23i)16-s + (0.331 + 0.573i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.755 + 0.655i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572191 - 1.53273i\)
\(L(\frac12)\) \(\approx\) \(0.572191 - 1.53273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - T \)
13 \( 1 + (-3.59 - 0.232i)T \)
good2 \( 1 + (-1.36 + 2.36i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (0.866 + 1.5i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.36 - 2.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.73 - 6.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.36 + 5.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.46T + 31T^{2} \)
37 \( 1 + (5.19 - 9i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.63 + 6.29i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.598 + 1.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + 2.53T + 53T^{2} \)
59 \( 1 + (2.83 + 4.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.69 - 13.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.598 - 1.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.633 - 1.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73T + 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + (4.36 - 7.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.59 - 4.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.98162997421715015239795622739, −11.28752560995680720658426065989, −10.21438197138394544086701684743, −9.921627800006830445413681924986, −8.603390932594624818858022422659, −6.22346768117121403045275735169, −5.47417495401766186800803169318, −4.02901896840208229107736067188, −3.33032416961313508940299034771, −1.38176906633482476661311864734, 3.11580129395178037719358741179, 4.77746623556523935925062039256, 5.68759122044395370545909048966, 6.59667619928395449414703329936, 7.33144503469277607537118619975, 8.593937219206852006091147951902, 9.404743442113346322941552398547, 11.37109497535048415253955850421, 12.47882981537166903256519954048, 13.04243327971250475792839699374

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