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Properties

Label 2-1232-77.20-c0-0-0
Degree $2$
Conductor $1232$
Sign $0.964 + 0.265i$
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  + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + 1.61·23-s + (−0.809 + 0.587i)25-s + (−0.5 − 0.363i)29-s + (1.30 + 0.951i)37-s − 0.618·43-s + (0.309 + 0.951i)49-s + (0.190 − 0.587i)53-s + (0.809 − 0.587i)63-s − 0.618·67-s + (−0.190 − 0.587i)71-s + (0.309 − 0.951i)77-s + (−0.190 + 0.587i)79-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + 1.61·23-s + (−0.809 + 0.587i)25-s + (−0.5 − 0.363i)29-s + (1.30 + 0.951i)37-s − 0.618·43-s + (0.309 + 0.951i)49-s + (0.190 − 0.587i)53-s + (0.809 − 0.587i)63-s − 0.618·67-s + (−0.190 − 0.587i)71-s + (0.309 − 0.951i)77-s + (−0.190 + 0.587i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ 0.964 + 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.168875178\)
\(L(\frac12)\) \(\approx\) \(1.168875178\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.618T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
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

−9.715708429310470801215819936593, −9.059262505943313902437925191320, −8.316721074757241251064680646202, −7.51202570295625856054842419987, −6.48464846713364718473076730065, −5.67793691137486277400329464511, −4.84392893199080262301725766528, −3.70157548404495699206300376854, −2.72099764306374197918307625911, −1.26563810431600125708071843424, 1.53854040227730921481415342678, 2.58950472534529719761011927287, 4.10400001206323126466211764423, 4.77256884090935572531406690547, 5.55278788370498961833474498219, 6.91499702183935283184843071122, 7.52190645971739551660270890448, 8.127992126134514133137730847553, 9.181480722890905046462997737048, 10.05356259877216911801332055347

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