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Properties

Label 2-1232-77.48-c0-0-0
Degree $2$
Conductor $1232$
Sign $0.331 + 0.943i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
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  + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s − 0.618·23-s + (0.309 − 0.951i)25-s + (−0.5 − 1.53i)29-s + (0.190 + 0.587i)37-s + 1.61·43-s + (−0.809 + 0.587i)49-s + (1.30 + 0.951i)53-s + (−0.309 + 0.951i)63-s + 1.61·67-s + (−1.30 + 0.951i)71-s + (−0.809 − 0.587i)77-s + (−1.30 − 0.951i)79-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s − 0.618·23-s + (0.309 − 0.951i)25-s + (−0.5 − 1.53i)29-s + (0.190 + 0.587i)37-s + 1.61·43-s + (−0.809 + 0.587i)49-s + (1.30 + 0.951i)53-s + (−0.309 + 0.951i)63-s + 1.61·67-s + (−1.30 + 0.951i)71-s + (−0.809 − 0.587i)77-s + (−1.30 − 0.951i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.331 + 0.943i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ 0.331 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9356373190\)
\(L(\frac12)\) \(\approx\) \(0.9356373190\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - 1.61T + T^{2} \)
71 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
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

−9.750781626655264011563172450751, −8.956751243674123114133593872781, −8.181706738589778989050067795795, −7.26621156211003848682512437511, −6.30748084129389519889889979420, −5.83341835388384917405537368685, −4.34157952251865343471472497318, −3.72020678993692712824848195598, −2.59591746557874268477463706604, −0.840032564715966728345163630414, 1.83203263244261924111559821474, 2.86058795416256148059944636157, 3.95118965721499041563046368770, 5.19020497070818483116764411460, 5.76460682695326382548566634948, 6.77063157495418659267574525537, 7.61036480406423776757074906690, 8.688267697508170169299354448265, 9.098841537073345523034217915280, 9.962640097704109491503909370614

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