Properties

Label 2-1232-1232.139-c0-0-1
Degree $2$
Conductor $1232$
Sign $0.461 + 0.886i$
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  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.587 − 0.809i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s − 18-s + (0.951 + 0.309i)22-s + 0.618·23-s + (0.951 + 0.309i)25-s + i·28-s + (−0.142 + 0.278i)29-s + i·32-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.587 − 0.809i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s − 18-s + (0.951 + 0.309i)22-s + 0.618·23-s + (0.951 + 0.309i)25-s + i·28-s + (−0.142 + 0.278i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.461 + 0.886i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ 0.461 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8696191048\)
\(L(\frac12)\) \(\approx\) \(0.8696191048\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (-0.587 + 0.809i)T^{2} \)
5 \( 1 + (-0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.587 - 0.809i)T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.412 + 0.809i)T + (-0.587 - 0.809i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.221 + 0.221i)T + iT^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.587 - 0.809i)T^{2} \)
61 \( 1 + (-0.951 - 0.309i)T^{2} \)
67 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
71 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
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

−9.892133781887678324406652424948, −9.039576099601195083197613489448, −8.298486901505559175747104190833, −7.41014669612434352237077394414, −6.87280662863395828967984567957, −5.25391967191222915493546155765, −4.46717402257202262098908928124, −3.51814801174464559447257241150, −2.30873762560547988739480324761, −1.15483492307582695350088833166, 1.38508010196545064757140024048, 2.64905461904732717544060496647, 4.47553083521713394457158767204, 5.06141650716783122140649810129, 5.87203676626175057478610839502, 6.94137601844431376222509016605, 7.78482059590272674787691229809, 8.243804352976102730709522775275, 9.035954489643240916315619090903, 10.00012532719180120383757391770

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