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Properties

Label 2-1232-7.2-c1-0-25
Degree $2$
Conductor $1232$
Sign $0.832 - 0.553i$
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  + (1.32 + 2.29i)3-s + (0.822 − 1.42i)5-s + (1.32 − 2.29i)7-s + (−2 + 3.46i)9-s + (0.5 + 0.866i)11-s + 5·13-s + 4.35·15-s + (−3 − 5.19i)17-s + (−2.82 + 4.88i)19-s + 7·21-s + (0.822 − 1.42i)23-s + (1.14 + 1.98i)25-s − 2.64·27-s + 6.29·29-s + (−2 − 3.46i)31-s + ⋯
L(s)  = 1  + (0.763 + 1.32i)3-s + (0.368 − 0.637i)5-s + (0.499 − 0.866i)7-s + (−0.666 + 1.15i)9-s + (0.150 + 0.261i)11-s + 1.38·13-s + 1.12·15-s + (−0.727 − 1.26i)17-s + (−0.647 + 1.12i)19-s + 1.52·21-s + (0.171 − 0.297i)23-s + (0.229 + 0.396i)25-s − 0.509·27-s + 1.16·29-s + (−0.359 − 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.539139951\)
\(L(\frac12)\) \(\approx\) \(2.539139951\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.29i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-1.32 - 2.29i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.822 + 1.42i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.82 - 4.88i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.822 + 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.29T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.82 - 3.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-1.35 + 2.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.822 + 1.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.32 - 4.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.14 - 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.96 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.35T + 71T^{2} \)
73 \( 1 + (0.177 + 0.306i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.32 - 2.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 + (3.29 - 5.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−9.679961168938911629421111227488, −9.000334426107062188247261961310, −8.486400991286372388260023788014, −7.57266150133322128631843363122, −6.39806735423414835507748121802, −5.27862558072729570742165936858, −4.36104354397071741211538853652, −3.98123321716685295986276572219, −2.74630877004431434470935594187, −1.27106890409501366008179539481, 1.33300736076809306614898087650, 2.28168011392371954220428266392, 3.02041335989670202357042529529, 4.34861408024812125919275887923, 5.87966854366577801474212495871, 6.37592617205597800754875725509, 7.10029650772126690964409702061, 8.235060347860093569773623670116, 8.587244754922997571226201536483, 9.241034675371293300883017886649

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