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Properties

Label 2-232-232.109-c1-0-6
Degree $2$
Conductor $232$
Sign $0.101 - 0.994i$
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  + (−1.35 + 0.391i)2-s + (2.95 + 1.42i)3-s + (1.69 − 1.06i)4-s + (−2.43 + 0.555i)5-s + (−4.57 − 0.776i)6-s + (−0.123 − 0.0594i)7-s + (−1.88 + 2.10i)8-s + (4.84 + 6.06i)9-s + (3.08 − 1.70i)10-s + (−2.20 + 2.76i)11-s + (6.51 − 0.735i)12-s + (3.18 + 2.53i)13-s + (0.191 + 0.0324i)14-s + (−7.98 − 1.82i)15-s + (1.73 − 3.60i)16-s − 3.76i·17-s + ⋯
L(s)  = 1  + (−0.960 + 0.276i)2-s + (1.70 + 0.821i)3-s + (0.846 − 0.532i)4-s + (−1.08 + 0.248i)5-s + (−1.86 − 0.317i)6-s + (−0.0466 − 0.0224i)7-s + (−0.666 + 0.745i)8-s + (1.61 + 2.02i)9-s + (0.976 − 0.539i)10-s + (−0.663 + 0.832i)11-s + (1.88 − 0.212i)12-s + (0.882 + 0.704i)13-s + (0.0510 + 0.00867i)14-s + (−2.06 − 0.470i)15-s + (0.433 − 0.901i)16-s − 0.912i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.101 - 0.994i$
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: \((2,\ 232,\ (\ :1/2),\ 0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877817 + 0.792537i\)
\(L(\frac12)\) \(\approx\) \(0.877817 + 0.792537i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.391i)T \)
29 \( 1 + (-5.29 - 0.958i)T \)
good3 \( 1 + (-2.95 - 1.42i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (2.43 - 0.555i)T + (4.50 - 2.16i)T^{2} \)
7 \( 1 + (0.123 + 0.0594i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (2.20 - 2.76i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-3.18 - 2.53i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + 3.76iT - 17T^{2} \)
19 \( 1 + (-2.61 + 1.25i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-1.18 + 5.17i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (3.56 - 0.813i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (2.82 + 3.54i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 12.3iT - 41T^{2} \)
43 \( 1 + (-1.00 + 4.41i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-6.02 - 4.80i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (7.89 - 1.80i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 1.03iT - 59T^{2} \)
61 \( 1 + (-1.11 - 0.536i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (1.68 - 1.34i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (1.30 - 1.63i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (2.02 + 0.462i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (-9.96 + 7.94i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-1.90 - 3.95i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-14.1 + 3.23i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (0.110 + 0.229i)T + (-60.4 + 75.8i)T^{2} \)
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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

−12.27346747516835841090923490691, −10.99263559897327259346132678156, −10.27633235950782269377521393738, −9.210182645193601541924991224540, −8.649412279389176965291455283115, −7.64274259888631201914922406315, −7.08506405357934771038977862199, −4.82188523843002828630948804735, −3.54498579244987851026935023542, −2.34322236506487701508205835752, 1.25965740304933866673216377352, 3.02296416785559205877396392282, 3.66134263491666210302504606842, 6.33659229612742869563744273163, 7.74670513109256103403406363289, 7.999288761158621072275102526340, 8.677111558170990474777405334399, 9.728400191193053112449209836683, 10.95756128093605888080601740827, 12.02522624318727177016125453265

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