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Properties

Label 2-232-29.9-c1-0-7
Degree $2$
Conductor $232$
Sign $0.362 + 0.932i$
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  + (2.26 − 1.80i)3-s + (−0.608 + 0.293i)5-s + (−2.74 − 3.44i)7-s + (1.19 − 5.22i)9-s + (1.48 − 0.338i)11-s + (1.33 + 5.84i)13-s + (−0.847 + 1.75i)15-s − 0.785i·17-s + (3.81 + 3.04i)19-s + (−12.4 − 2.83i)21-s + (3.68 + 1.77i)23-s + (−2.83 + 3.55i)25-s + (−2.96 − 6.15i)27-s + (−1.91 − 5.03i)29-s + (1.58 + 3.28i)31-s + ⋯
L(s)  = 1  + (1.30 − 1.04i)3-s + (−0.272 + 0.131i)5-s + (−1.03 − 1.30i)7-s + (0.397 − 1.74i)9-s + (0.447 − 0.102i)11-s + (0.370 + 1.62i)13-s + (−0.218 + 0.454i)15-s − 0.190i·17-s + (0.874 + 0.697i)19-s + (−2.70 − 0.618i)21-s + (0.767 + 0.369i)23-s + (−0.566 + 0.710i)25-s + (−0.570 − 1.18i)27-s + (−0.354 − 0.934i)29-s + (0.283 + 0.589i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.362 + 0.932i$
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: \((2,\ 232,\ (\ :1/2),\ 0.362 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34812 - 0.922279i\)
\(L(\frac12)\) \(\approx\) \(1.34812 - 0.922279i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.91 + 5.03i)T \)
good3 \( 1 + (-2.26 + 1.80i)T + (0.667 - 2.92i)T^{2} \)
5 \( 1 + (0.608 - 0.293i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (2.74 + 3.44i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-1.48 + 0.338i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-1.33 - 5.84i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 0.785iT - 17T^{2} \)
19 \( 1 + (-3.81 - 3.04i)T + (4.22 + 18.5i)T^{2} \)
23 \( 1 + (-3.68 - 1.77i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-1.58 - 3.28i)T + (-19.3 + 24.2i)T^{2} \)
37 \( 1 + (-2.57 - 0.586i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 - 0.0403iT - 41T^{2} \)
43 \( 1 + (-4.73 + 9.82i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (7.69 - 1.75i)T + (42.3 - 20.3i)T^{2} \)
53 \( 1 + (12.7 - 6.16i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + 8.32T + 59T^{2} \)
61 \( 1 + (-5.11 + 4.07i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (1.07 - 4.72i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (2.40 + 10.5i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.42 - 2.96i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (-4.39 - 1.00i)T + (71.1 + 34.2i)T^{2} \)
83 \( 1 + (7.19 - 9.02i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-5.15 - 10.6i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 + (5.10 + 4.07i)T + (21.5 + 94.5i)T^{2} \)
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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

−12.21769962034862115554905127944, −11.18405481007653327388307548376, −9.678429686856579574527932875783, −9.145973701340976994107802477943, −7.81410575282541753432607982913, −7.12667381355769090251162514768, −6.40874423468460379959747367299, −4.01388684871589745846880664770, −3.22906002947265380855706566761, −1.45970805686764242198390066067, 2.78337714508613019396299277026, 3.39412245572383203487187388621, 4.87983262452194707516382432096, 6.12890213654671593701777439561, 7.80057276909118979528891266968, 8.683536898505744280512116509761, 9.390481611927741804563211060322, 10.07392469841355900147211324231, 11.30434848660971243637629980434, 12.62912312972578867547774871230

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