↑

Properties

Label 2-132-132.59-c1-0-11
Degree $2$
Conductor $132$
Sign $0.956 - 0.291i$
Analytic cond. $1.05402$
Root an. cond. $1.02665$
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  + (1.39 − 0.221i)2-s + (0.0877 + 1.72i)3-s + (1.90 − 0.618i)4-s + (−0.951 − 1.30i)5-s + (0.505 + 2.39i)6-s + (0.224 − 0.0729i)7-s + (2.52 − 1.28i)8-s + (−2.98 + 0.303i)9-s + (−1.61 − 1.61i)10-s + (−2.19 + 2.48i)11-s + (1.23 + 3.23i)12-s + (0.809 + 0.587i)13-s + (0.297 − 0.151i)14-s + (2.18 − 1.76i)15-s + (3.23 − 2.35i)16-s + (−2.99 − 4.11i)17-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (0.0506 + 0.998i)3-s + (0.951 − 0.309i)4-s + (−0.425 − 0.585i)5-s + (0.206 + 0.978i)6-s + (0.0848 − 0.0275i)7-s + (0.891 − 0.453i)8-s + (−0.994 + 0.101i)9-s + (−0.511 − 0.511i)10-s + (−0.660 + 0.750i)11-s + (0.356 + 0.934i)12-s + (0.224 + 0.163i)13-s + (0.0795 − 0.0405i)14-s + (0.563 − 0.454i)15-s + (0.809 − 0.587i)16-s + (−0.725 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.956 - 0.291i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.956 - 0.291i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70588 + 0.253747i\)
\(L(\frac12)\) \(\approx\) \(1.70588 + 0.253747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.221i)T \)
3 \( 1 + (-0.0877 - 1.72i)T \)
11 \( 1 + (2.19 - 2.48i)T \)
good5 \( 1 + (0.951 + 1.30i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.224 + 0.0729i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.99 + 4.11i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.61 + 1.5i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.76T + 23T^{2} \)
29 \( 1 + (-3.80 + 1.23i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.44 - 4.73i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.16 - 9.73i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-9.00 - 2.92i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.23iT - 43T^{2} \)
47 \( 1 + (-2.04 + 6.29i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.96 - 9.59i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.263 + 0.812i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.92 + 3.57i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.85iT - 67T^{2} \)
71 \( 1 + (-4.92 + 3.57i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.38 + 7.33i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.0 + 13.8i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.66 - 2.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 7.18iT - 89T^{2} \)
97 \( 1 + (2.69 + 1.95i)T + (29.9 + 92.2i)T^{2} \)
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

−13.31249078351047952768702110787, −12.35670313784406338475242753092, −11.32210002952245037453731020036, −10.51122069259890010294843400101, −9.310246366279591863146133043299, −8.006228393400400820544339933977, −6.48021708422892600312119209074, −4.88064725372968424259749656099, −4.44330333240931049320174090524, −2.75880974266261361520967380925, 2.37348215418277050622438245048, 3.78014885572110743643503760751, 5.63485458601818150814195633123, 6.55934273935704737928632765487, 7.63193462783083624893261803192, 8.524861463151820323856471768874, 10.87773685285099866187661903684, 11.18021395365841751571125918954, 12.65549405075574518545466739232, 13.03413310849736106456056693977

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