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Properties

Label 2-460-5.4-c3-0-31
Degree $2$
Conductor $460$
Sign $-0.646 - 0.763i$
Analytic cond. $27.1408$
Root an. cond. $5.20969$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.27i·3-s + (8.53 − 7.22i)5-s − 19.6i·7-s − 41.5·9-s − 49.7·11-s + 2.13i·13-s + (−59.8 − 70.6i)15-s + 122. i·17-s − 66.2·19-s − 162.·21-s − 23i·23-s + (20.5 − 123. i)25-s + 120. i·27-s + 153.·29-s − 161.·31-s + ⋯
L(s)  = 1  − 1.59i·3-s + (0.763 − 0.646i)5-s − 1.06i·7-s − 1.53·9-s − 1.36·11-s + 0.0455i·13-s + (−1.02 − 1.21i)15-s + 1.75i·17-s − 0.799·19-s − 1.69·21-s − 0.208i·23-s + (0.164 − 0.986i)25-s + 0.856i·27-s + 0.981·29-s − 0.936·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.646 - 0.763i$
Analytic conductor: \(27.1408\)
Root analytic conductor: \(5.20969\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :3/2),\ -0.646 - 0.763i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.111222612\)
\(L(\frac12)\) \(\approx\) \(1.111222612\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.53 + 7.22i)T \)
23 \( 1 + 23iT \)
good3 \( 1 + 8.27iT - 27T^{2} \)
7 \( 1 + 19.6iT - 343T^{2} \)
11 \( 1 + 49.7T + 1.33e3T^{2} \)
13 \( 1 - 2.13iT - 2.19e3T^{2} \)
17 \( 1 - 122. iT - 4.91e3T^{2} \)
19 \( 1 + 66.2T + 6.85e3T^{2} \)
29 \( 1 - 153.T + 2.43e4T^{2} \)
31 \( 1 + 161.T + 2.97e4T^{2} \)
37 \( 1 + 324. iT - 5.06e4T^{2} \)
41 \( 1 - 342.T + 6.89e4T^{2} \)
43 \( 1 + 7.56iT - 7.95e4T^{2} \)
47 \( 1 + 315. iT - 1.03e5T^{2} \)
53 \( 1 - 177. iT - 1.48e5T^{2} \)
59 \( 1 + 129.T + 2.05e5T^{2} \)
61 \( 1 + 514.T + 2.26e5T^{2} \)
67 \( 1 - 807. iT - 3.00e5T^{2} \)
71 \( 1 + 948.T + 3.57e5T^{2} \)
73 \( 1 + 364. iT - 3.89e5T^{2} \)
79 \( 1 - 415.T + 4.93e5T^{2} \)
83 \( 1 - 157. iT - 5.71e5T^{2} \)
89 \( 1 - 1.26e3T + 7.04e5T^{2} \)
97 \( 1 - 234. iT - 9.12e5T^{2} \)
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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

−10.37521677209429822275730513948, −8.879415646747705240716065932659, −8.066025539504740056156950116813, −7.37617291200243285729893260050, −6.34044812687088986070268454245, −5.61013231608380570688239860493, −4.21442870244203012706472419060, −2.43877611265185644681031339166, −1.49952637684896396038640230285, −0.33517841246090756515909226844, 2.49589201859109096323706917018, 3.08765451184905004346728286440, 4.73707356781785155898473489952, 5.32368493198967653390407344597, 6.25334224149969925286877302875, 7.68684813115011556620601505080, 8.949461938147928378163862611950, 9.474365352157261400705599988980, 10.32002998914294254375326978741, 10.88285691513465330112698467383

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