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Properties

Label 2-1232-1232.1091-c0-0-0
Degree $2$
Conductor $1232$
Sign $-0.362 - 0.932i$
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 − 1.61·23-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 − 1.61·23-s + (0.587 + 0.809i)25-s + (−0.587 − 0.809i)28-s + (−1.76 + 0.278i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.362 - 0.932i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ -0.362 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2061431689\)
\(L(\frac12)\) \(\approx\) \(0.2061431689\)
\(L(1)\) 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 \( 1 + 1.61T + T^{2} \)
29 \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.0489i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.412 + 0.809i)T + (-0.587 + 0.809i)T^{2} \)
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 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.887602938155762086205093544614, −9.589430835063544826449191369457, −8.587016015623536326399098602250, −8.073034049543974624497767984219, −7.04827140405308661989642855590, −5.91747252287111999078081009094, −5.10077150507869618460355570691, −3.71710172664198280976364136967, −2.89812461634642437353597433554, −1.98883326653225428069550130028, 0.19918730950394005512864721361, 2.16673473614840386751393645106, 3.55363502459720291426686689560, 4.74378564642983972287435459562, 5.72163705590477266034716072995, 6.30201113327226470759172006542, 7.38921108148169569028760859489, 7.958954192105439015106108901341, 8.697598379135617225918434017980, 9.626292805955175185552870367875

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