↑

Properties

Label 2-26e2-52.35-c0-0-0
Degree $2$
Conductor $676$
Sign $0.0128 + 0.999i$
Analytic cond. $0.337367$
Root an. cond. $0.580833$
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 + 5-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 0.999·18-s + (−0.499 − 0.866i)20-s + (0.5 − 0.866i)29-s + (0.499 + 0.866i)32-s + 0.999·34-s + (−0.499 + 0.866i)36-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 5-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 0.999·18-s + (−0.499 − 0.866i)20-s + (0.5 − 0.866i)29-s + (0.499 + 0.866i)32-s + 0.999·34-s + (−0.499 + 0.866i)36-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(0.337367\)
Root analytic conductor: \(0.580833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :0),\ 0.0128 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.249391756\)
\(L(\frac12)\) \(\approx\) \(1.249391756\)
\(L(1)\) 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 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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

−10.40612840938130986801918374269, −9.819646071846981292767651557812, −9.103850340188820011764627955841, −8.139341141101000658600779560963, −6.42951895461459833645524950158, −5.99563493144545861227856720640, −4.97380682300253742096406869782, −3.75618254366669232592181114926, −2.74088633300785962010973814662, −1.46887780319472174272838567176, 2.25209354893256740392722789970, 3.44431808183015820872253462935, 4.96117934701175018273544896232, 5.41825727822592713064704476063, 6.39316653007563906906795756891, 7.31095056935532996142984383035, 8.209414007866471659017570627123, 9.076181747713083303278403858072, 9.890754419320423622699847712530, 10.90305454732790366866465704632

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