↑

Properties

Label 2-2132-2132.1151-c0-0-0
Degree $2$
Conductor $2132$
Sign $-0.951 - 0.306i$
Analytic cond. $1.06400$
Root an. cond. $1.03150$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 1.93i·5-s + 0.999·8-s + (−0.258 + 0.965i)9-s + (−1.67 − 0.965i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (1.57 + 1.20i)17-s + (−0.707 − 0.707i)18-s + (1.67 − 0.965i)20-s − 2.73·25-s + (0.499 + 0.866i)26-s + (0.965 + 1.25i)29-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 1.93i·5-s + 0.999·8-s + (−0.258 + 0.965i)9-s + (−1.67 − 0.965i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (1.57 + 1.20i)17-s + (−0.707 − 0.707i)18-s + (1.67 − 0.965i)20-s − 2.73·25-s + (0.499 + 0.866i)26-s + (0.965 + 1.25i)29-s + (−0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2132\)    =    \(2^{2} \cdot 13 \cdot 41\)
Sign: $-0.951 - 0.306i$
Analytic conductor: \(1.06400\)
Root analytic conductor: \(1.03150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2132} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2132,\ (\ :0),\ -0.951 - 0.306i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8399446584\)
\(L(\frac12)\) \(\approx\) \(0.8399446584\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + T \)
good3 \( 1 + (0.258 - 0.965i)T^{2} \)
5 \( 1 - 1.93iT - T^{2} \)
7 \( 1 + (0.258 + 0.965i)T^{2} \)
11 \( 1 + (0.965 + 0.258i)T^{2} \)
17 \( 1 + (-1.57 - 1.20i)T + (0.258 + 0.965i)T^{2} \)
19 \( 1 + (-0.965 + 0.258i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.965 - 1.25i)T + (-0.258 + 0.965i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.707 - 0.707i)T^{2} \)
53 \( 1 + (1.46 + 0.607i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.258 - 0.965i)T^{2} \)
71 \( 1 + (-0.258 - 0.965i)T^{2} \)
73 \( 1 - 0.517iT - T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (1.83 + 0.241i)T + (0.965 + 0.258i)T^{2} \)
97 \( 1 + (-0.0999 - 0.758i)T + (-0.965 + 0.258i)T^{2} \)
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

−9.903348203665539377321959644228, −8.436123443302773597085526054802, −8.062945241034408049762501072105, −7.27365146216349579315146980004, −6.65497992692013769145871076790, −5.81055279742837600705947454424, −5.28067588119394509089020331135, −3.78231429523232027557391779683, −3.00068394603086615431668472481, −1.73483537369139536910725347000, 0.77291886177314735981547667994, 1.54282607618878356031978299394, 3.01469604746826536048079094070, 3.99128617798024937859067503081, 4.71372191997613107539498950076, 5.50561164384503337099580041228, 6.62305564333640496734384748398, 7.87009543945473194141320403302, 8.326970597933381027009182155374, 9.136255509124433539456825832993

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