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Properties

Label 2-1232-1232.797-c0-0-1
Degree $2$
Conductor $1232$
Sign $-0.0327 - 0.999i$
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 + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (−0.951 + 0.309i)9-s + (0.951 + 0.309i)11-s − 14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)18-s + (−0.809 + 0.587i)22-s − 1.17i·23-s + (−0.587 + 0.809i)25-s + (0.587 − 0.809i)28-s + (1.76 + 0.278i)29-s − i·32-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (−0.951 + 0.309i)9-s + (0.951 + 0.309i)11-s − 14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)18-s + (−0.809 + 0.587i)22-s − 1.17i·23-s + (−0.587 + 0.809i)25-s + (0.587 − 0.809i)28-s + (1.76 + 0.278i)29-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0327 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0327 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.0327 - 0.999i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ -0.0327 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7849408268\)
\(L(\frac12)\) \(\approx\) \(0.7849408268\)
\(L(1)\) 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 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-0.951 - 0.309i)T \)
good3 \( 1 + (0.951 - 0.309i)T^{2} \)
5 \( 1 + (0.587 - 0.809i)T^{2} \)
13 \( 1 + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.951 - 0.309i)T^{2} \)
23 \( 1 + 1.17iT - T^{2} \)
29 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 - 1.95i)T + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (0.642 - 0.642i)T - iT^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-1.58 - 0.809i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.951 + 0.309i)T^{2} \)
61 \( 1 + (-0.587 + 0.809i)T^{2} \)
67 \( 1 + (0.642 + 0.642i)T + iT^{2} \)
71 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.951863320414553755352659962710, −8.984714855853478588339467788339, −8.543832814425282029284907252633, −7.87695104658254477434472018910, −6.73527685337606396615619168959, −6.12050137293231667356526707391, −5.18334572630188991707868942048, −4.47009318623725027278486792632, −2.80191704974724335127454403165, −1.52274930273656096415461315139, 0.935578760010153649293124548995, 2.26794066716906862739771615198, 3.53714842143786951315290288037, 4.15260156020788372259864950486, 5.38142701314334636763637143025, 6.57268463661192841925744897503, 7.45209746652859566004153596592, 8.350913469207235823342432335189, 8.856439318783594994507113851140, 9.798030862999802434887924611362

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