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Properties

Label 2-232-232.109-c1-0-0
Degree $2$
Conductor $232$
Sign $-0.919 + 0.392i$
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  + (1.17 + 0.789i)2-s + (−2.17 − 1.04i)3-s + (0.752 + 1.85i)4-s + (−2.51 + 0.574i)5-s + (−1.72 − 2.95i)6-s + (−4.18 − 2.01i)7-s + (−0.581 + 2.76i)8-s + (1.77 + 2.22i)9-s + (−3.40 − 1.31i)10-s + (1.61 − 2.02i)11-s + (0.305 − 4.82i)12-s + (−2.33 − 1.85i)13-s + (−3.31 − 5.67i)14-s + (6.08 + 1.38i)15-s + (−2.86 + 2.78i)16-s + 7.17i·17-s + ⋯
L(s)  = 1  + (0.829 + 0.558i)2-s + (−1.25 − 0.605i)3-s + (0.376 + 0.926i)4-s + (−1.12 + 0.257i)5-s + (−0.705 − 1.20i)6-s + (−1.58 − 0.762i)7-s + (−0.205 + 0.978i)8-s + (0.592 + 0.742i)9-s + (−1.07 − 0.415i)10-s + (0.487 − 0.610i)11-s + (0.0883 − 1.39i)12-s + (−0.646 − 0.515i)13-s + (−0.887 − 1.51i)14-s + (1.57 + 0.358i)15-s + (−0.717 + 0.696i)16-s + 1.73i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $-0.919 + 0.392i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ -0.919 + 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000703974 - 0.00344269i\)
\(L(\frac12)\) \(\approx\) \(0.000703974 - 0.00344269i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 0.789i)T \)
29 \( 1 + (-1.04 + 5.28i)T \)
good3 \( 1 + (2.17 + 1.04i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (2.51 - 0.574i)T + (4.50 - 2.16i)T^{2} \)
7 \( 1 + (4.18 + 2.01i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-1.61 + 2.02i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.33 + 1.85i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 7.17iT - 17T^{2} \)
19 \( 1 + (2.71 - 1.30i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.662 + 2.90i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (0.338 - 0.0773i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-4.05 - 5.08i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 8.13iT - 41T^{2} \)
43 \( 1 + (0.228 - 0.999i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (7.17 + 5.72i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (10.9 - 2.49i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + 2.42iT - 59T^{2} \)
61 \( 1 + (-4.89 - 2.35i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (3.92 - 3.13i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (0.274 - 0.344i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (9.79 + 2.23i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (6.46 - 5.15i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (2.57 + 5.35i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-4.44 + 1.01i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (4.73 + 9.83i)T + (-60.4 + 75.8i)T^{2} \)
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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

−12.70163526620909222702516937865, −12.04857466763390424962227538447, −11.13864928534807360699906365874, −10.24793633397711501545202726191, −8.344869552841102898726835962930, −7.31348083088993423158982977511, −6.46550788633004863670627708826, −5.96383828953826799611410647561, −4.26331630645550196587145975876, −3.35671313253870072877224658689, 0.00241668338767143270253631473, 2.98232301333115059264283467421, 4.30531341613510055026944888433, 5.08516224320705119443780198613, 6.29156444563545619910560865557, 7.10762841350377157401758654424, 9.339811129262465967283203530655, 9.779912636245851471387267150582, 11.12440676822779573814482180887, 11.77618671217586874739655961150

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