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Properties

Label 2-1232-1232.69-c0-0-1
Degree $2$
Conductor $1232$
Sign $0.502 + 0.864i$
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 − 1.90i·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 − 1.90i·23-s + (−0.951 + 0.309i)25-s − i·28-s + (−0.142 − 0.278i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.502 + 0.864i$
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: \((2,\ 1232,\ (\ :0),\ 0.502 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.557275596\)
\(L(\frac12)\) \(\approx\) \(1.557275596\)
\(L(1)\) 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 \( 1 + 1.90iT - T^{2} \)
29 \( 1 + (0.142 + 0.278i)T + (-0.587 + 0.809i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (1.58 - 0.809i)T + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \)
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 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
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

−10.14886810652797299370512982203, −9.033271877666909319689691107689, −8.346040307073846827514832640241, −7.23122890644318760644302835493, −6.32423990800433885616982431245, −5.17342542608059533697977057152, −4.62886752649630479521445540959, −3.76472819505186534413445357301, −2.29472723787291273595806906189, −1.60368664987761114517879135136, 1.58612876081511881577815378676, 3.44262651396047723971752374448, 3.97789733036738363915412658922, 5.03049323897806166873824965250, 5.86768624516877817346448628513, 6.75764565203599952825224784624, 7.46971488506705883292448188973, 8.247066733098554892401944870380, 9.094758924386923090049047433061, 9.801310177776018653806681034555

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