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Properties

Label 2-460-23.12-c1-0-6
Degree $2$
Conductor $460$
Sign $0.264 + 0.964i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (0.147 − 1.02i)3-s + (0.959 − 0.281i)5-s + (−0.636 − 0.734i)7-s + (1.84 + 0.540i)9-s + (−4.49 − 2.88i)11-s + (3.87 − 4.47i)13-s + (−0.147 − 1.02i)15-s + (1.45 + 3.18i)17-s + (1.27 − 2.79i)19-s + (−0.850 + 0.546i)21-s + (0.107 + 4.79i)23-s + (0.841 − 0.540i)25-s + (2.12 − 4.65i)27-s + (−4.38 − 9.60i)29-s + (0.341 + 2.37i)31-s + ⋯
L(s)  = 1  + (0.0854 − 0.594i)3-s + (0.429 − 0.125i)5-s + (−0.240 − 0.277i)7-s + (0.613 + 0.180i)9-s + (−1.35 − 0.870i)11-s + (1.07 − 1.24i)13-s + (−0.0382 − 0.265i)15-s + (0.353 + 0.773i)17-s + (0.293 − 0.642i)19-s + (−0.185 + 0.119i)21-s + (0.0224 + 0.999i)23-s + (0.168 − 0.108i)25-s + (0.408 − 0.895i)27-s + (−0.814 − 1.78i)29-s + (0.0612 + 0.426i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.264 + 0.964i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.264 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19380 - 0.910609i\)
\(L(\frac12)\) \(\approx\) \(1.19380 - 0.910609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (-0.107 - 4.79i)T \)
good3 \( 1 + (-0.147 + 1.02i)T + (-2.87 - 0.845i)T^{2} \)
7 \( 1 + (0.636 + 0.734i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (4.49 + 2.88i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-3.87 + 4.47i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-1.45 - 3.18i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-1.27 + 2.79i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (4.38 + 9.60i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.341 - 2.37i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (5.02 + 1.47i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-4.51 + 1.32i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (0.336 - 2.34i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 5.26T + 47T^{2} \)
53 \( 1 + (-8.54 - 9.86i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (2.73 - 3.15i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (-1.72 - 11.9i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (5.08 - 3.27i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (10.5 - 6.75i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (2.92 - 6.40i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-10.7 + 12.4i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-4.70 - 1.38i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (2.10 - 14.6i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-2.25 + 0.661i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.55984937261848837781172953527, −10.31808936996282641712363369932, −8.956100627230224251960295582765, −7.969341529329803436653610594313, −7.40901044272519204313601136749, −6.01107015591472703090585383655, −5.46607185548003919174009181331, −3.84204322974902291129111763922, −2.60750933204653686403715020619, −1.00406993099286466988974370735, 1.87845167489689912320503980085, 3.31329233758319351696480600547, 4.51568436238989229656108317230, 5.43305725045996859756349420445, 6.64849994112483180264348104585, 7.49259726644943116828798484889, 8.809683964557188062242970187741, 9.511478131733207611333167804056, 10.30194481938439389107611515374, 10.97144479435047173767796452950

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