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Properties

Label 2-232-29.13-c1-0-3
Degree $2$
Conductor $232$
Sign $0.535 + 0.844i$
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  + (−1.79 − 1.42i)3-s + (2.68 + 1.29i)5-s + (0.839 − 1.05i)7-s + (0.499 + 2.18i)9-s + (−0.512 − 0.116i)11-s + (1.27 − 5.57i)13-s + (−2.95 − 6.13i)15-s + 0.733i·17-s + (4.33 − 3.45i)19-s + (−3.00 + 0.686i)21-s + (3.93 − 1.89i)23-s + (2.40 + 3.01i)25-s + (−0.750 + 1.55i)27-s + (−2.92 − 4.52i)29-s + (−4.11 + 8.53i)31-s + ⋯
L(s)  = 1  + (−1.03 − 0.824i)3-s + (1.19 + 0.577i)5-s + (0.317 − 0.398i)7-s + (0.166 + 0.729i)9-s + (−0.154 − 0.0352i)11-s + (0.353 − 1.54i)13-s + (−0.763 − 1.58i)15-s + 0.177i·17-s + (0.994 − 0.792i)19-s + (−0.656 + 0.149i)21-s + (0.820 − 0.395i)23-s + (0.480 + 0.602i)25-s + (−0.144 + 0.300i)27-s + (−0.542 − 0.839i)29-s + (−0.738 + 1.53i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.960677 - 0.528199i\)
\(L(\frac12)\) \(\approx\) \(0.960677 - 0.528199i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.92 + 4.52i)T \)
good3 \( 1 + (1.79 + 1.42i)T + (0.667 + 2.92i)T^{2} \)
5 \( 1 + (-2.68 - 1.29i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-0.839 + 1.05i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (0.512 + 0.116i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-1.27 + 5.57i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 0.733iT - 17T^{2} \)
19 \( 1 + (-4.33 + 3.45i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (-3.93 + 1.89i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (4.11 - 8.53i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (4.04 - 0.922i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 - 9.24iT - 41T^{2} \)
43 \( 1 + (-0.964 - 2.00i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-4.20 - 0.959i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-11.0 - 5.30i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 4.67T + 59T^{2} \)
61 \( 1 + (3.08 + 2.45i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (3.01 + 13.1i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (3.04 - 13.3i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.74 + 9.85i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (1.53 - 0.351i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (-2.75 - 3.45i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (0.626 - 1.30i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (5.80 - 4.62i)T + (21.5 - 94.5i)T^{2} \)
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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

−12.07171941113971437528882051685, −10.88735223912665068992216063644, −10.52059680124220214349240408548, −9.252926476542392719560588395549, −7.74548045916875340268934900279, −6.83285107617751233760681420438, −5.89485655438262379956166058969, −5.16578095106293660984648296521, −2.94534380533999275029416957054, −1.18604060875981256702943910598, 1.82777634160958960680078444572, 4.08241064344036836155025863764, 5.41095799370873172958799333071, 5.64743588099238626759677991103, 7.11734475724435645671278000078, 8.891953702746352571625147236930, 9.470657536032609082109341014490, 10.41043247693401323420288023154, 11.39498733697511441465615347517, 12.06982439263761313634587264674

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