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Properties

Label 2-232-29.9-c1-0-6
Degree $2$
Conductor $232$
Sign $0.783 + 0.621i$
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  + (1.96 − 1.56i)3-s + (1.24 − 0.599i)5-s + (1.40 + 1.76i)7-s + (0.737 − 3.23i)9-s + (−5.31 + 1.21i)11-s + (0.381 + 1.67i)13-s + (1.50 − 3.12i)15-s − 2.80i·17-s + (−1.12 − 0.900i)19-s + (5.51 + 1.25i)21-s + (−4.13 − 1.98i)23-s + (−1.92 + 2.41i)25-s + (−0.342 − 0.712i)27-s + (4.19 + 3.38i)29-s + (−3.39 − 7.05i)31-s + ⋯
L(s)  = 1  + (1.13 − 0.904i)3-s + (0.556 − 0.268i)5-s + (0.530 + 0.665i)7-s + (0.245 − 1.07i)9-s + (−1.60 + 0.366i)11-s + (0.105 + 0.463i)13-s + (0.389 − 0.807i)15-s − 0.679i·17-s + (−0.259 − 0.206i)19-s + (1.20 + 0.274i)21-s + (−0.861 − 0.414i)23-s + (−0.385 + 0.483i)25-s + (−0.0659 − 0.137i)27-s + (0.778 + 0.627i)29-s + (−0.609 − 1.26i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69039 - 0.589064i\)
\(L(\frac12)\) \(\approx\) \(1.69039 - 0.589064i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.19 - 3.38i)T \)
good3 \( 1 + (-1.96 + 1.56i)T + (0.667 - 2.92i)T^{2} \)
5 \( 1 + (-1.24 + 0.599i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-1.40 - 1.76i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (5.31 - 1.21i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.381 - 1.67i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 2.80iT - 17T^{2} \)
19 \( 1 + (1.12 + 0.900i)T + (4.22 + 18.5i)T^{2} \)
23 \( 1 + (4.13 + 1.98i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (3.39 + 7.05i)T + (-19.3 + 24.2i)T^{2} \)
37 \( 1 + (-8.65 - 1.97i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + 2.09iT - 41T^{2} \)
43 \( 1 + (3.86 - 8.01i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (-6.00 + 1.37i)T + (42.3 - 20.3i)T^{2} \)
53 \( 1 + (5.71 - 2.75i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + (-2.35 + 1.87i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (2.57 - 11.2i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-0.688 - 3.01i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (0.476 - 0.990i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (-10.4 - 2.38i)T + (71.1 + 34.2i)T^{2} \)
83 \( 1 + (-10.8 + 13.5i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (5.20 + 10.8i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 + (7.18 + 5.72i)T + (21.5 + 94.5i)T^{2} \)
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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

−12.41047034555760828591134171798, −11.24059542171534952265157748287, −9.905082856862809996662272918172, −8.980367433294545597818958240386, −8.085393182081825331035192138738, −7.41307573976622500490793564624, −5.97991495145800097322617719163, −4.76802731147259018837114471181, −2.75129601951167541872708400354, −1.94012396575922587690256223784, 2.34335806273503966816839081067, 3.54738999562432495654940039454, 4.74827415250909438599691613822, 6.02541439907089902401222935527, 7.80885230229232986405753990213, 8.230496717472029758889397362009, 9.503226677676864928026849900997, 10.45289894680183581282978697096, 10.71995843222004718044416926782, 12.46902641715233091369934978581

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