↑

Properties

Label 2-195-13.4-c1-0-6
Degree $2$
Conductor $195$
Sign $0.752 - 0.658i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.88 + 1.08i)2-s + (0.5 − 0.866i)3-s + (1.36 + 2.36i)4-s + i·5-s + (1.88 − 1.08i)6-s + (−0.383 + 0.221i)7-s + 1.59i·8-s + (−0.499 − 0.866i)9-s + (−1.08 + 1.88i)10-s + (−1.37 − 0.796i)11-s + 2.73·12-s + (−3.39 − 1.20i)13-s − 0.964·14-s + (0.866 + 0.5i)15-s + (0.999 − 1.73i)16-s + (1.88 + 3.26i)17-s + ⋯
L(s)  = 1  + (1.33 + 0.769i)2-s + (0.288 − 0.499i)3-s + (0.683 + 1.18i)4-s + 0.447i·5-s + (0.769 − 0.444i)6-s + (−0.145 + 0.0837i)7-s + 0.563i·8-s + (−0.166 − 0.288i)9-s + (−0.343 + 0.595i)10-s + (−0.415 − 0.240i)11-s + 0.788·12-s + (−0.942 − 0.335i)13-s − 0.257·14-s + (0.223 + 0.129i)15-s + (0.249 − 0.433i)16-s + (0.456 + 0.791i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.752 - 0.658i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16266 + 0.811822i\)
\(L(\frac12)\) \(\approx\) \(2.16266 + 0.811822i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (3.39 + 1.20i)T \)
good2 \( 1 + (-1.88 - 1.08i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (0.383 - 0.221i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.37 + 0.796i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.88 - 3.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.33 - 0.773i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.352 + 0.611i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.24 - 2.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.18iT - 31T^{2} \)
37 \( 1 + (8.96 + 5.17i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.21 - 3.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.36 - 4.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.41iT - 47T^{2} \)
53 \( 1 - 5.51T + 53T^{2} \)
59 \( 1 + (-7.55 + 4.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.86 - 8.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.8 + 6.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.1 + 7.57i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 9.12iT - 83T^{2} \)
89 \( 1 + (4.11 + 2.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.28 - 4.20i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.75798402104225249352336707446, −12.24967474186664475493149052758, −10.87376244317476727221200626447, −9.640359837216512782966653528057, −8.065448935259834057931155469522, −7.28870658360863331354620249755, −6.25284260236353547386612674020, −5.35277836078121305133720118502, −3.90996190086605052071664079011, −2.67222331081325700964309964649, 2.29055920048751160040757445839, 3.55490633192227674880971428354, 4.74981156038661236163395564576, 5.40373684346054508126269951034, 7.08263733269721505981995793132, 8.534910524928263220173167558983, 9.758191767001936189851393650129, 10.58448987568548792383901059159, 11.76171552857705870240566953217, 12.37174907470707619088471808354

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