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Properties

Label 2-132-33.26-c2-0-1
Degree $2$
Conductor $132$
Sign $-0.262 - 0.964i$
Analytic cond. $3.59673$
Root an. cond. $1.89650$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.296 + 2.98i)3-s + (3.94 + 5.43i)5-s + (−0.935 − 2.87i)7-s + (−8.82 − 1.76i)9-s + (−6.22 + 9.07i)11-s + (−0.130 − 0.0946i)13-s + (−17.3 + 10.1i)15-s + (14.3 + 19.7i)17-s + (−0.192 + 0.591i)19-s + (8.87 − 1.93i)21-s − 6.09i·23-s + (−6.20 + 19.0i)25-s + (7.89 − 25.8i)27-s + (49.6 − 16.1i)29-s + (−36.1 − 26.2i)31-s + ⋯
L(s)  = 1  + (−0.0987 + 0.995i)3-s + (0.789 + 1.08i)5-s + (−0.133 − 0.411i)7-s + (−0.980 − 0.196i)9-s + (−0.565 + 0.824i)11-s + (−0.0100 − 0.00727i)13-s + (−1.15 + 0.678i)15-s + (0.844 + 1.16i)17-s + (−0.0101 + 0.0311i)19-s + (0.422 − 0.0923i)21-s − 0.264i·23-s + (−0.248 + 0.763i)25-s + (0.292 − 0.956i)27-s + (1.71 − 0.556i)29-s + (−1.16 − 0.846i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $-0.262 - 0.964i$
Analytic conductor: \(3.59673\)
Root analytic conductor: \(1.89650\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1),\ -0.262 - 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.840560 + 1.10020i\)
\(L(\frac12)\) \(\approx\) \(0.840560 + 1.10020i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.296 - 2.98i)T \)
11 \( 1 + (6.22 - 9.07i)T \)
good5 \( 1 + (-3.94 - 5.43i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (0.935 + 2.87i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (0.130 + 0.0946i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (-14.3 - 19.7i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (0.192 - 0.591i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + 6.09iT - 529T^{2} \)
29 \( 1 + (-49.6 + 16.1i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (36.1 + 26.2i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-9.51 - 29.2i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (1.80 + 0.585i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 79.5T + 1.84e3T^{2} \)
47 \( 1 + (14.0 + 4.56i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-28.0 + 38.6i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (20.8 - 6.77i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-49.8 + 36.2i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 56.3T + 4.48e3T^{2} \)
71 \( 1 + (-52.9 - 72.9i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (21.7 + 67.0i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-79.8 - 58.0i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (60.8 + 83.7i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 165. iT - 7.92e3T^{2} \)
97 \( 1 + (79.1 + 57.5i)T + (2.90e3 + 8.94e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.54387360379321922078795492717, −12.25597623841167465926833690300, −10.83996723354535522651844837303, −10.27741575810260521117610021233, −9.605285009357391168562561779203, −8.044078224046244674187177733569, −6.62458953907386190094190906097, −5.54674427870091436119981662593, −4.04230014897372180267666753136, −2.59680062002240595783735851813, 1.01777200930924035325154420164, 2.74089716873212650622260900913, 5.20204218924125664321511218846, 5.88166356105158184996872205082, 7.35757487962635547014293860825, 8.559787651350014237949117419103, 9.320655351604491658703826116379, 10.79856002134497587257120702781, 12.10327872608005785643970910746, 12.68685167891488160307992302315

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