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Properties

Label 2-1232-7.2-c1-0-27
Degree $2$
Conductor $1232$
Sign $0.827 + 0.561i$
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  + (0.207 + 0.358i)3-s + (1.70 − 2.95i)5-s + (2.62 + 0.358i)7-s + (1.41 − 2.44i)9-s + (0.5 + 0.866i)11-s + 1.82·13-s + 1.41·15-s + (3.82 + 6.63i)17-s + (−1.70 + 2.95i)19-s + (0.414 + 1.01i)21-s + (1.12 − 1.94i)23-s + (−3.32 − 5.76i)25-s + 2.41·27-s − 8.65·29-s + (−2 − 3.46i)31-s + ⋯
L(s)  = 1  + (0.119 + 0.207i)3-s + (0.763 − 1.32i)5-s + (0.990 + 0.135i)7-s + (0.471 − 0.816i)9-s + (0.150 + 0.261i)11-s + 0.507·13-s + 0.365·15-s + (0.928 + 1.60i)17-s + (−0.391 + 0.678i)19-s + (0.0903 + 0.221i)21-s + (0.233 − 0.404i)23-s + (−0.665 − 1.15i)25-s + 0.464·27-s − 1.60·29-s + (−0.359 − 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.827 + 0.561i$
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: \((2,\ 1232,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.376107887\)
\(L(\frac12)\) \(\approx\) \(2.376107887\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.358i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.207 - 0.358i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.82T + 13T^{2} \)
17 \( 1 + (-3.82 - 6.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.70 - 2.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.12 + 1.94i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.29 + 5.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.58T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + (-3.24 + 5.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.94 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.20 + 7.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.08 - 5.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.62 - 9.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.07T + 71T^{2} \)
73 \( 1 + (-3.29 - 5.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.37 - 4.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.1T + 83T^{2} \)
89 \( 1 + (-2.24 + 3.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.82T + 97T^{2} \)
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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

−9.584039072585232145715305126255, −8.746323332983983580631913752874, −8.330809868366710708438986213658, −7.28452743964094573402695282383, −5.90082833782051260858773696020, −5.59671282808120311801566003721, −4.36505882253948151849299092970, −3.80502141466561097451213847574, −1.89662910988810988491518488172, −1.23187416074648973047204664094, 1.48669507545229224292878013407, 2.47584334545804387681865554557, 3.43582504747756555725779251957, 4.83677143227443591637514099787, 5.54039280383235118055395296516, 6.65912576543505764352879724812, 7.31928789493241657808952580672, 7.935134415795430969629338323096, 9.067677356937518852724875729093, 9.887158846617842178881359337986

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