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Properties

Label 2-1232-1232.69-c0-0-0
Degree $2$
Conductor $1232$
Sign $0.404 - 0.914i$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s + (0.587 + 0.809i)9-s + (−0.587 + 0.809i)11-s − 0.999·14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (0.309 − 0.951i)22-s + 1.90i·23-s + (−0.951 + 0.309i)25-s + (0.951 − 0.309i)28-s + (−0.896 − 1.76i)29-s + i·32-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s + (0.587 + 0.809i)9-s + (−0.587 + 0.809i)11-s − 0.999·14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (0.309 − 0.951i)22-s + 1.90i·23-s + (−0.951 + 0.309i)25-s + (0.951 − 0.309i)28-s + (−0.896 − 1.76i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.404 - 0.914i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ 0.404 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7556066957\)
\(L(\frac12)\) \(\approx\) \(0.7556066957\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (0.587 - 0.809i)T \)
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 \( 1 - 1.90iT - T^{2} \)
29 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.412i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (-1.95 - 0.309i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.587 + 0.809i)T^{2} \)
61 \( 1 + (-0.951 + 0.309i)T^{2} \)
67 \( 1 + (0.221 - 0.221i)T - iT^{2} \)
71 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
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

−9.860510539158909033765222351761, −9.399536295764221842572676999123, −8.226121318955728340208278250431, −7.63629407544720520588733262346, −7.26384757501122887570088303402, −5.80437481941307230185084554666, −5.26014443394376531874923024001, −4.13460464149589260173253432966, −2.34934910201172777502722168999, −1.67598102427497264381246574947, 0.957672539574581188681465192459, 2.24486620233356232575847598790, 3.45441499815305769486528988732, 4.43406281446202420293034258303, 5.72115158778706196497246050524, 6.72163712576001157222715759066, 7.45054196318353566314355829549, 8.354198899814730089101067709114, 8.796673134293485362351631326872, 9.855869196217677636647686745441

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