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Properties

Label 2-1232-7.2-c1-0-6
Degree $2$
Conductor $1232$
Sign $-0.991 - 0.126i$
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  + (1.5 + 2.59i)3-s + (−1 + 1.73i)5-s + (2.5 − 0.866i)7-s + (−3 + 5.19i)9-s + (−0.5 − 0.866i)11-s − 7·13-s − 6·15-s + (−1 − 1.73i)17-s + (6 + 5.19i)21-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s − 9·27-s − 5·29-s + (2 + 3.46i)31-s + (1.5 − 2.59i)33-s + ⋯
L(s)  = 1  + (0.866 + 1.49i)3-s + (−0.447 + 0.774i)5-s + (0.944 − 0.327i)7-s + (−1 + 1.73i)9-s + (−0.150 − 0.261i)11-s − 1.94·13-s − 1.54·15-s + (−0.242 − 0.420i)17-s + (1.30 + 1.13i)21-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s − 1.73·27-s − 0.928·29-s + (0.359 + 0.622i)31-s + (0.261 − 0.452i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.991 - 0.126i$
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: \((2,\ 1232,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.568919532\)
\(L(\frac12)\) \(\approx\) \(1.568919532\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 7T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−10.03146454469973941843336117471, −9.443847355288976217760005088400, −8.586209261995286975396502043715, −7.55829359733811636773785731528, −7.33976850984693552569103797080, −5.53883738560711113978460943248, −4.79219075851345261069699665288, −4.03249304927294422839988843204, −3.12934629850793228752000032895, −2.24542729203734979377731293547, 0.55880597618365303799358078163, 2.05469976422295651603303473795, 2.46725093493603377212467699267, 4.14940286886637377538873032831, 4.99281726169508142518167465232, 6.12881203801496026509970315939, 7.23514055539152128717146384807, 7.72182592534246826609921992607, 8.342934949165103955203798872746, 8.966184762355098303523003181164

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