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Properties

Label 2-1232-1232.83-c0-0-0
Degree $2$
Conductor $1232$
Sign $0.402 + 0.915i$
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 + (0.142 − 0.896i)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 + (0.142 − 0.896i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.541482897\)
\(L(\frac12)\) \(\approx\) \(1.541482897\)
\(L(1)\) 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 \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \)
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 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
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

−9.645062351840774725294944604795, −9.378206016406711591673217093533, −8.279462689006299964231314630951, −7.20357421437792630985612448341, −6.15276113201367770199626365920, −5.49417764768395636520431027474, −4.29362289661801830540484769536, −3.85742134174793540403881000511, −2.34103924855175528111519650644, −1.51219323732943436630657710882, 1.64139503830711887335128928925, 3.33403040407212813186324175584, 4.32521060889179090648659959537, 4.70754351472726037189496953919, 6.09546764952673332375978644182, 6.63437394848706720654650362733, 7.69364613439666769638267724055, 8.019861315740882245316211582067, 9.127855882649693252400826581103, 10.00606262640775949325939477341

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