↑

Properties

Label 2-195-3.2-c2-0-26
Degree $2$
Conductor $195$
Sign $-0.865 + 0.500i$
Analytic cond. $5.31336$
Root an. cond. $2.30507$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.62i·2-s + (−1.50 − 2.59i)3-s + 1.34·4-s − 2.23i·5-s + (−4.22 + 2.44i)6-s + 9.21·7-s − 8.70i·8-s + (−4.49 + 7.79i)9-s − 3.64·10-s − 18.7i·11-s + (−2.02 − 3.50i)12-s + 3.60·13-s − 15.0i·14-s + (−5.80 + 3.35i)15-s − 8.78·16-s + 30.5i·17-s + ⋯
L(s)  = 1  − 0.814i·2-s + (−0.500 − 0.865i)3-s + 0.337·4-s − 0.447i·5-s + (−0.704 + 0.407i)6-s + 1.31·7-s − 1.08i·8-s + (−0.499 + 0.866i)9-s − 0.364·10-s − 1.70i·11-s + (−0.168 − 0.291i)12-s + 0.277·13-s − 1.07i·14-s + (−0.387 + 0.223i)15-s − 0.549·16-s + 1.79i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.865 + 0.500i$
Analytic conductor: \(5.31336\)
Root analytic conductor: \(2.30507\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1),\ -0.865 + 0.500i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.408708 - 1.52413i\)
\(L(\frac12)\) \(\approx\) \(0.408708 - 1.52413i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.50 + 2.59i)T \)
5 \( 1 + 2.23iT \)
13 \( 1 - 3.60T \)
good2 \( 1 + 1.62iT - 4T^{2} \)
7 \( 1 - 9.21T + 49T^{2} \)
11 \( 1 + 18.7iT - 121T^{2} \)
17 \( 1 - 30.5iT - 289T^{2} \)
19 \( 1 + 27.6T + 361T^{2} \)
23 \( 1 - 10.6iT - 529T^{2} \)
29 \( 1 + 13.5iT - 841T^{2} \)
31 \( 1 - 11.3T + 961T^{2} \)
37 \( 1 - 5.25T + 1.36e3T^{2} \)
41 \( 1 - 15.5iT - 1.68e3T^{2} \)
43 \( 1 - 27.7T + 1.84e3T^{2} \)
47 \( 1 + 55.7iT - 2.20e3T^{2} \)
53 \( 1 - 23.0iT - 2.80e3T^{2} \)
59 \( 1 - 32.7iT - 3.48e3T^{2} \)
61 \( 1 - 90.7T + 3.72e3T^{2} \)
67 \( 1 - 131.T + 4.48e3T^{2} \)
71 \( 1 - 21.4iT - 5.04e3T^{2} \)
73 \( 1 - 48.5T + 5.32e3T^{2} \)
79 \( 1 + 74.7T + 6.24e3T^{2} \)
83 \( 1 - 25.8iT - 6.88e3T^{2} \)
89 \( 1 - 89.9iT - 7.92e3T^{2} \)
97 \( 1 + 47.6T + 9.40e3T^{2} \)
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

−11.67752900240498084086058451024, −11.10413703713779776690569600338, −10.50096285006694673469053895870, −8.524008191468081237115209244265, −8.068534473750973961668347770849, −6.49105359662495980762415537176, −5.63069643623568288853743318783, −3.96639718580804914207564348051, −2.12927441297088328773865091434, −1.03298771436608612024218165721, 2.32000015958910573968498879994, 4.47803222213311779875738167492, 5.17245384885734666811103096637, 6.54511741111478759180142530736, 7.38766648219018984319080725584, 8.540400969262221817397668155826, 9.797649260161649716422925080348, 10.88589373594789508562747487341, 11.44421745286723539235618981260, 12.42064826502181558415836737828

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