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Properties

Label 2-1232-308.263-c1-0-28
Degree $2$
Conductor $1232$
Sign $0.987 + 0.155i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.31 + 0.758i)3-s + (−0.222 − 0.385i)5-s + (2.33 + 1.23i)7-s + (−0.350 − 0.607i)9-s + (−1.27 − 3.06i)11-s − 3.68i·13-s − 0.674i·15-s + (1.86 + 1.07i)17-s + (2.83 + 4.91i)19-s + (2.13 + 3.39i)21-s + (7.10 − 4.10i)23-s + (2.40 − 4.15i)25-s − 5.61i·27-s + 4.09i·29-s + (−3.31 − 1.91i)31-s + ⋯
L(s)  = 1  + (0.758 + 0.437i)3-s + (−0.0994 − 0.172i)5-s + (0.884 + 0.466i)7-s + (−0.116 − 0.202i)9-s + (−0.383 − 0.923i)11-s − 1.02i·13-s − 0.174i·15-s + (0.452 + 0.261i)17-s + (0.651 + 1.12i)19-s + (0.466 + 0.740i)21-s + (1.48 − 0.854i)23-s + (0.480 − 0.831i)25-s − 1.07i·27-s + 0.760i·29-s + (−0.594 − 0.343i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.987 + 0.155i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.987 + 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.306275456\)
\(L(\frac12)\) \(\approx\) \(2.306275456\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.33 - 1.23i)T \)
11 \( 1 + (1.27 + 3.06i)T \)
good3 \( 1 + (-1.31 - 0.758i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.222 + 0.385i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.68iT - 13T^{2} \)
17 \( 1 + (-1.86 - 1.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.83 - 4.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.10 + 4.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.09iT - 29T^{2} \)
31 \( 1 + (3.31 + 1.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.522 - 0.904i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.32iT - 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (1.79 - 1.03i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.30 - 9.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.4 + 6.59i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.3 - 5.96i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.50 - 0.871i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.53iT - 71T^{2} \)
73 \( 1 + (-7.15 - 4.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.17 + 3.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + (-2.62 - 4.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−9.481339816728978354339354084477, −8.819686400828755968951536521557, −8.147900105855529171958033439657, −7.66193841797370703323807204895, −6.15117562870500604671844882564, −5.45368712708081567431773838653, −4.51085618775062608239747388700, −3.31993304816455060925606041466, −2.72837735986363626748092019930, −1.05707207241876203193752192368, 1.39067190804962941539388720164, 2.38253876833676074269812103489, 3.42307834186326263824727350094, 4.71264788598254379949667477157, 5.23696875486992584846082217400, 6.82547661210968457062273099458, 7.47415376628695412439548534487, 7.80137295459879436952447787588, 9.143188827164521790213377046765, 9.309804346619614337333063408805

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