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Properties

Label 2-195-65.4-c1-0-1
Degree $2$
Conductor $195$
Sign $-0.701 - 0.712i$
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  + (−0.611 + 1.05i)2-s + (0.866 + 0.5i)3-s + (0.251 + 0.436i)4-s + (−2.22 + 0.247i)5-s + (−1.05 + 0.611i)6-s + (0.997 + 1.72i)7-s − 3.06·8-s + (0.499 + 0.866i)9-s + (1.09 − 2.50i)10-s + (0.0539 + 0.0311i)11-s + 0.503i·12-s + (−1.38 + 3.33i)13-s − 2.44·14-s + (−2.04 − 0.896i)15-s + (1.36 − 2.37i)16-s + (5.17 − 2.99i)17-s + ⋯
L(s)  = 1  + (−0.432 + 0.749i)2-s + (0.499 + 0.288i)3-s + (0.125 + 0.218i)4-s + (−0.993 + 0.110i)5-s + (−0.432 + 0.249i)6-s + (0.377 + 0.653i)7-s − 1.08·8-s + (0.166 + 0.288i)9-s + (0.346 − 0.792i)10-s + (0.0162 + 0.00938i)11-s + 0.145i·12-s + (−0.383 + 0.923i)13-s − 0.652·14-s + (−0.528 − 0.231i)15-s + (0.342 − 0.592i)16-s + (1.25 − 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.701 - 0.712i$
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: \((2,\ 195,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373086 + 0.891226i\)
\(L(\frac12)\) \(\approx\) \(0.373086 + 0.891226i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.247i)T \)
13 \( 1 + (1.38 - 3.33i)T \)
good2 \( 1 + (0.611 - 1.05i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.997 - 1.72i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0539 - 0.0311i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-5.17 + 2.99i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.14 - 0.661i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.713 - 0.411i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.04 - 7.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.04iT - 31T^{2} \)
37 \( 1 + (-2.72 + 4.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.94 - 5.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.52 + 4.34i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.45T + 47T^{2} \)
53 \( 1 + 1.39iT - 53T^{2} \)
59 \( 1 + (3.05 - 1.76i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.83 + 6.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.17 - 8.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.394 - 0.227i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.99T + 73T^{2} \)
79 \( 1 + 8.01T + 79T^{2} \)
83 \( 1 + 4.14T + 83T^{2} \)
89 \( 1 + (9.96 + 5.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.91 + 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.62229345350899386348508014900, −11.92307451519424996204244791646, −11.04627328309084884629241730713, −9.440304208620424272370446471946, −8.771164515766095520723595361338, −7.71460471521372023016806379970, −7.18778807884870940220059690860, −5.62467930248968899101946787390, −4.09659555922774819921998328231, −2.75781308025619985690381666797, 0.993739744697694019342143568523, 2.83871822958921233084317673594, 4.10611512930894987477986831842, 5.82521261285140587039603913057, 7.41427434110711114135080557685, 8.065392402836277483076483231774, 9.247198964509382921300883684594, 10.37397314387147337782368084419, 11.01765656656288988341499628316, 12.13727875141861625549716681504

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