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Properties

Label 2-195-65.4-c1-0-4
Degree $2$
Conductor $195$
Sign $0.711 - 0.702i$
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  + (−0.173 + 0.300i)2-s + (−0.866 − 0.5i)3-s + (0.939 + 1.62i)4-s + (−0.956 − 2.02i)5-s + (0.300 − 0.173i)6-s + (2.09 + 3.62i)7-s − 1.34·8-s + (0.499 + 0.866i)9-s + (0.774 + 0.0633i)10-s + (4.15 + 2.39i)11-s − 1.87i·12-s + (2.38 − 2.70i)13-s − 1.45·14-s + (−0.182 + 2.22i)15-s + (−1.64 + 2.84i)16-s + (0.309 − 0.178i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.212i)2-s + (−0.499 − 0.288i)3-s + (0.469 + 0.813i)4-s + (−0.427 − 0.903i)5-s + (0.122 − 0.0709i)6-s + (0.790 + 1.36i)7-s − 0.476·8-s + (0.166 + 0.288i)9-s + (0.244 + 0.0200i)10-s + (1.25 + 0.723i)11-s − 0.542i·12-s + (0.661 − 0.750i)13-s − 0.388·14-s + (−0.0470 + 0.575i)15-s + (−0.411 + 0.712i)16-s + (0.0750 − 0.0433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.711 - 0.702i$
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: \((2,\ 195,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00839 + 0.414125i\)
\(L(\frac12)\) \(\approx\) \(1.00839 + 0.414125i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.956 + 2.02i)T \)
13 \( 1 + (-2.38 + 2.70i)T \)
good2 \( 1 + (0.173 - 0.300i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-2.09 - 3.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.15 - 2.39i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.309 + 0.178i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.15 - 1.82i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.49 - 1.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.583 - 1.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.22iT - 31T^{2} \)
37 \( 1 + (-3.42 + 5.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.54 + 5.51i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.05 + 1.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 - 0.595iT - 53T^{2} \)
59 \( 1 + (-1.79 + 1.03i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.16 - 5.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.80 + 3.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.05 + 4.65i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.473T + 73T^{2} \)
79 \( 1 - 5.50T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (1.35 + 0.782i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.36 + 7.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.39467246555566103679241244007, −11.82238643057012081448622874608, −11.16126165490313756203746377210, −9.296392752696664730666791978317, −8.472943408636094541496579099434, −7.72500103635129343195499310566, −6.38355950587661854803723323760, −5.30775446079015962235594099916, −3.92128901925024851505590505410, −1.90856266244527814277790918609, 1.28731295851903355619209454128, 3.54747280685622436534376233671, 4.72631625628082737414796613552, 6.55808178384016042060683231780, 6.73249440003016535801408184460, 8.389639138654464775656046849444, 9.778838785106611093139684243378, 10.78715641497759786513568634511, 11.15952404070457109351434039431, 11.79239717556735092132603626414

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