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Properties

Label 2-1188-33.14-c0-0-1
Degree $2$
Conductor $1188$
Sign $0.794 + 0.606i$
Analytic cond. $0.592889$
Root an. cond. $0.769993$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.363 − 0.5i)5-s + (0.309 − 0.951i)7-s + (0.587 + 0.809i)11-s + (0.5 − 0.363i)13-s + (−0.951 + 1.30i)17-s + (−0.309 − 0.951i)19-s − 1.61i·23-s + (0.190 + 0.587i)25-s + (−0.951 − 0.309i)29-s + (0.809 − 0.587i)31-s + (−0.363 − 0.5i)35-s + (0.951 − 0.309i)41-s + (−0.587 + 0.190i)47-s + (0.587 + 0.809i)53-s + 0.618·55-s + ⋯
L(s)  = 1  + (0.363 − 0.5i)5-s + (0.309 − 0.951i)7-s + (0.587 + 0.809i)11-s + (0.5 − 0.363i)13-s + (−0.951 + 1.30i)17-s + (−0.309 − 0.951i)19-s − 1.61i·23-s + (0.190 + 0.587i)25-s + (−0.951 − 0.309i)29-s + (0.809 − 0.587i)31-s + (−0.363 − 0.5i)35-s + (0.951 − 0.309i)41-s + (−0.587 + 0.190i)47-s + (0.587 + 0.809i)53-s + 0.618·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1188\)    =    \(2^{2} \cdot 3^{3} \cdot 11\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(0.592889\)
Root analytic conductor: \(0.769993\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1188} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1188,\ (\ :0),\ 0.794 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.174330702\)
\(L(\frac12)\) \(\approx\) \(1.174330702\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (-0.587 - 0.809i)T \)
good5 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + 1.61iT - T^{2} \)
29 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + 0.618iT - T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−9.917397591031840893747590099467, −8.959620480002244121977232494802, −8.403491417386630226166959411255, −7.32584530524934686889299808483, −6.60679364363528776037687084801, −5.70358311961819130596329944897, −4.37165140458756664098657322499, −4.16775171144960481679170870218, −2.44008596032532827160683901066, −1.22481052008835227657992357793, 1.70001876331664119263397658104, 2.80246892806892132727784614608, 3.82693249166291112335362106074, 5.06691205027073138789889716664, 5.93472840939171981027857102385, 6.56437425616165049932101646110, 7.57539918254544364004133421684, 8.632228616669339121409512063709, 9.111687999236528839000120377292, 9.953622530733981062465938465564

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