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Properties

Label 2-232-29.4-c1-0-4
Degree $2$
Conductor $232$
Sign $0.736 + 0.676i$
Analytic cond. $1.85252$
Root an. cond. $1.36107$
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  + (−0.661 − 1.37i)3-s + (−0.510 + 2.23i)5-s + (3.30 − 1.59i)7-s + (0.423 − 0.530i)9-s + (1.90 − 1.51i)11-s + (−1.02 − 1.28i)13-s + (3.40 − 0.777i)15-s − 0.984i·17-s + (2.07 − 4.30i)19-s + (−4.36 − 3.48i)21-s + (0.990 + 4.33i)23-s + (−0.229 − 0.110i)25-s + (−5.46 − 1.24i)27-s + (−3.99 + 3.61i)29-s + (0.835 + 0.190i)31-s + ⋯
L(s)  = 1  + (−0.381 − 0.792i)3-s + (−0.228 + 0.999i)5-s + (1.24 − 0.601i)7-s + (0.141 − 0.176i)9-s + (0.573 − 0.457i)11-s + (−0.284 − 0.356i)13-s + (0.879 − 0.200i)15-s − 0.238i·17-s + (0.476 − 0.988i)19-s + (−0.953 − 0.760i)21-s + (0.206 + 0.904i)23-s + (−0.0458 − 0.0220i)25-s + (−1.05 − 0.240i)27-s + (−0.741 + 0.671i)29-s + (0.149 + 0.0342i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.736 + 0.676i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12943 - 0.439969i\)
\(L(\frac12)\) \(\approx\) \(1.12943 - 0.439969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (3.99 - 3.61i)T \)
good3 \( 1 + (0.661 + 1.37i)T + (-1.87 + 2.34i)T^{2} \)
5 \( 1 + (0.510 - 2.23i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-3.30 + 1.59i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-1.90 + 1.51i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.02 + 1.28i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 0.984iT - 17T^{2} \)
19 \( 1 + (-2.07 + 4.30i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.990 - 4.33i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (-0.835 - 0.190i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (-6.58 - 5.25i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 - 3.72iT - 41T^{2} \)
43 \( 1 + (8.73 - 1.99i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-6.29 + 5.01i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (3.04 - 13.3i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 6.44T + 59T^{2} \)
61 \( 1 + (3.25 + 6.76i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (3.24 - 4.06i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (7.83 + 9.81i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (15.7 - 3.58i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (6.11 + 4.87i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (-2.11 - 1.01i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-11.2 - 2.56i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (-5.69 + 11.8i)T + (-60.4 - 75.8i)T^{2} \)
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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

−11.68761181344480411444398112522, −11.41436151969253390061639235630, −10.43954888794584505749126790507, −9.116255552551392764722747007362, −7.64173188998855940571222835370, −7.23780629403923394294198560690, −6.15180445780727517878127075120, −4.72869588033038029819556730260, −3.20964194660084010811916410152, −1.31440666869631439027689446203, 1.78527952969482084376102472123, 4.19446155346082234784380800951, 4.80915020679788468289962887657, 5.78269406800114027558635360241, 7.53366664823149249424183107646, 8.495178348105645301911210221261, 9.372053768302604266036754436528, 10.38973126150480132838765294706, 11.48298255817234526273399956806, 12.06977397655938174941642979515

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