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Properties

Label 2-232-29.22-c1-0-7
Degree $2$
Conductor $232$
Sign $-0.0509 + 0.998i$
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  + (1.42 − 2.95i)3-s + (−0.136 − 0.597i)5-s + (0.294 + 0.141i)7-s + (−4.84 − 6.07i)9-s + (3.71 + 2.96i)11-s + (−2.63 + 3.29i)13-s + (−1.96 − 0.447i)15-s − 1.72i·17-s + (−0.863 − 1.79i)19-s + (0.838 − 0.668i)21-s + (−1.12 + 4.93i)23-s + (4.16 − 2.00i)25-s + (−15.2 + 3.48i)27-s + (−5.38 + 0.218i)29-s + (8.88 − 2.02i)31-s + ⋯
L(s)  = 1  + (0.822 − 1.70i)3-s + (−0.0610 − 0.267i)5-s + (0.111 + 0.0535i)7-s + (−1.61 − 2.02i)9-s + (1.12 + 0.894i)11-s + (−0.729 + 0.914i)13-s + (−0.506 − 0.115i)15-s − 0.417i·17-s + (−0.197 − 0.411i)19-s + (0.183 − 0.145i)21-s + (−0.234 + 1.02i)23-s + (0.833 − 0.401i)25-s + (−2.93 + 0.671i)27-s + (−0.999 + 0.0406i)29-s + (1.59 − 0.364i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0509 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0509 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $-0.0509 + 0.998i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ -0.0509 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05987 - 1.11535i\)
\(L(\frac12)\) \(\approx\) \(1.05987 - 1.11535i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.38 - 0.218i)T \)
good3 \( 1 + (-1.42 + 2.95i)T + (-1.87 - 2.34i)T^{2} \)
5 \( 1 + (0.136 + 0.597i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (-0.294 - 0.141i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-3.71 - 2.96i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (2.63 - 3.29i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 1.72iT - 17T^{2} \)
19 \( 1 + (0.863 + 1.79i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (1.12 - 4.93i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-8.88 + 2.02i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-4.28 + 3.41i)T + (8.23 - 36.0i)T^{2} \)
41 \( 1 - 6.90iT - 41T^{2} \)
43 \( 1 + (-7.25 - 1.65i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (-2.73 - 2.18i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (-0.145 - 0.639i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + (6.65 - 13.8i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (-0.567 - 0.711i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (2.65 - 3.32i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (5.98 + 1.36i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (-3.37 + 2.69i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-6.50 + 3.13i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (4.11 - 0.938i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (0.123 + 0.257i)T + (-60.4 + 75.8i)T^{2} \)
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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

−12.04984826842405342952519680302, −11.55621809779097651424620837533, −9.518622608005384654572825795688, −8.986323792420303374070995682576, −7.75686866708489758374523319463, −7.08319545476393027599486291000, −6.21022015925441957960437754150, −4.36510362000461987446505357949, −2.63365464456037672889597336846, −1.42787007848821253361991268209, 2.81477367741165309377452810223, 3.79654214838431999126213545938, 4.83214766308898107230946498779, 6.11636749499473878049379041737, 7.87010801964068139927284804707, 8.714209486905743270602060486053, 9.533516434907515375404840748110, 10.49491089229557162730341892606, 11.03684959832557593653107990659, 12.34223720994703124861326579084

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