Properties

Label 2-460-460.99-c1-0-56
Degree $2$
Conductor $460$
Sign $-0.677 + 0.735i$
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  + (0.926 − 1.06i)2-s + (−1.23 + 0.791i)3-s + (−0.284 − 1.97i)4-s + (2.03 − 0.928i)5-s + (−0.294 + 2.04i)6-s + (−1.19 − 4.05i)7-s + (−2.37 − 1.52i)8-s + (−0.356 + 0.779i)9-s + (0.890 − 3.03i)10-s + (1.91 + 2.21i)12-s + (−5.44 − 2.48i)14-s + (−1.76 + 2.75i)15-s + (−3.83 + 1.12i)16-s + (0.503 + 1.10i)18-s + (−2.41 − 3.76i)20-s + (4.67 + 4.05i)21-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (−0.710 + 0.456i)3-s + (−0.142 − 0.989i)4-s + (0.909 − 0.415i)5-s + (−0.120 + 0.836i)6-s + (−0.450 − 1.53i)7-s + (−0.841 − 0.540i)8-s + (−0.118 + 0.259i)9-s + (0.281 − 0.959i)10-s + (0.553 + 0.638i)12-s + (−1.45 − 0.664i)14-s + (−0.456 + 0.710i)15-s + (−0.959 + 0.281i)16-s + (0.118 + 0.259i)18-s + (−0.540 − 0.841i)20-s + (1.02 + 0.884i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.677 + 0.735i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.677 + 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583190 - 1.32975i\)
\(L(\frac12)\) \(\approx\) \(0.583190 - 1.32975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.926 + 1.06i)T \)
5 \( 1 + (-2.03 + 0.928i)T \)
23 \( 1 + (3.12 + 3.64i)T \)
good3 \( 1 + (1.23 - 0.791i)T + (1.24 - 2.72i)T^{2} \)
7 \( 1 + (1.19 + 4.05i)T + (-5.88 + 3.78i)T^{2} \)
11 \( 1 + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-1.51 + 10.5i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-12.8 - 28.1i)T^{2} \)
37 \( 1 + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-4.25 - 9.32i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-7.05 - 10.9i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 - 4.51T + 47T^{2} \)
53 \( 1 + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-5.31 + 8.26i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (-8.31 - 7.20i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (10.1 + 70.2i)T^{2} \)
73 \( 1 + (70.0 - 20.5i)T^{2} \)
79 \( 1 + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (0.390 + 0.178i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (-8.78 - 13.6i)T + (-36.9 + 80.9i)T^{2} \)
97 \( 1 + (-63.5 + 73.3i)T^{2} \)
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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

−10.74387753941238388817687036520, −9.995531790192315092744272987749, −9.605823468382753786638851428121, −8.022979088903746823485450233836, −6.49808858959510026403790920959, −5.88313249533016640039301806393, −4.69439218647397281327605725322, −4.11508301396101979306133359551, −2.49163424101138036030790430734, −0.792712488146846992028172898318, 2.28430137264973192135951332770, 3.45886687300711931627994882836, 5.38876393305403323682116997414, 5.66930615794763347301227064775, 6.51199231141481470923992087078, 7.28309764252453696471891385449, 8.823676105670072054427297314007, 9.231782679617600893865918272320, 10.65127610244792692400742106702, 11.78201344169848019258719182407

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