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Properties

Label 2-132-44.19-c1-0-11
Degree $2$
Conductor $132$
Sign $-0.889 - 0.457i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.246 − 1.39i)2-s + (−0.951 + 0.309i)3-s + (−1.87 + 0.686i)4-s + (−3.06 − 2.22i)5-s + (0.664 + 1.24i)6-s + (−0.793 + 2.44i)7-s + (1.41 + 2.44i)8-s + (0.809 − 0.587i)9-s + (−2.34 + 4.81i)10-s + (−3.19 − 0.874i)11-s + (1.57 − 1.23i)12-s + (−2.76 − 3.79i)13-s + (3.59 + 0.503i)14-s + (3.60 + 1.17i)15-s + (3.05 − 2.57i)16-s + (0.685 − 0.943i)17-s + ⋯
L(s)  = 1  + (−0.174 − 0.984i)2-s + (−0.549 + 0.178i)3-s + (−0.939 + 0.343i)4-s + (−1.37 − 0.996i)5-s + (0.271 + 0.509i)6-s + (−0.299 + 0.922i)7-s + (0.501 + 0.865i)8-s + (0.269 − 0.195i)9-s + (−0.742 + 1.52i)10-s + (−0.964 − 0.263i)11-s + (0.454 − 0.355i)12-s + (−0.765 − 1.05i)13-s + (0.960 + 0.134i)14-s + (0.930 + 0.302i)15-s + (0.764 − 0.644i)16-s + (0.166 − 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $-0.889 - 0.457i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ -0.889 - 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0499008 + 0.206012i\)
\(L(\frac12)\) \(\approx\) \(0.0499008 + 0.206012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.246 + 1.39i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (3.19 + 0.874i)T \)
good5 \( 1 + (3.06 + 2.22i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.793 - 2.44i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.76 + 3.79i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.685 + 0.943i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.147 - 0.454i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.80iT - 23T^{2} \)
29 \( 1 + (-3.70 - 1.20i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.70 + 5.10i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.163 - 0.504i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.197 + 0.0640i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.387T + 43T^{2} \)
47 \( 1 + (11.6 - 3.76i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (8.74 - 6.35i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-5.25 - 1.70i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.33 + 8.72i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 6.06iT - 67T^{2} \)
71 \( 1 + (0.999 - 1.37i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.26 + 1.71i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.3 + 7.51i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.06 + 5.86i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.485T + 89T^{2} \)
97 \( 1 + (-2.60 + 1.89i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.53520424722275137235789810104, −11.79939614507300230923049403350, −10.83527207444985186966381327324, −9.694002529116262481459674171472, −8.509253454322047253382983494060, −7.77376608264338442886584324199, −5.44317636261988613894107773655, −4.57419746664920964378956798446, −3.00283708201777200077903115454, −0.24465774889430257241097700369, 3.72155718924778332983580392477, 4.94032305880803805452234438519, 6.71806778413133267659043775581, 7.23479230075028350346379856908, 8.074381326975375050993552206115, 9.837769687953115700735596353140, 10.70795529397457325511856709821, 11.76793549564047211670422137970, 12.97398213378899172374982327691, 14.10242055522133905426296453149

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