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Properties

Label 2-460-92.91-c1-0-14
Degree $2$
Conductor $460$
Sign $-0.969 - 0.246i$
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  + (0.949 + 1.04i)2-s + 2.47i·3-s + (−0.198 + 1.99i)4-s + i·5-s + (−2.59 + 2.35i)6-s + 1.26·7-s + (−2.27 + 1.68i)8-s − 3.14·9-s + (−1.04 + 0.949i)10-s + 3.32·11-s + (−4.93 − 0.491i)12-s + 1.65·13-s + (1.19 + 1.32i)14-s − 2.47·15-s + (−3.92 − 0.788i)16-s − 6.70i·17-s + ⋯
L(s)  = 1  + (0.671 + 0.741i)2-s + 1.43i·3-s + (−0.0990 + 0.995i)4-s + 0.447i·5-s + (−1.06 + 0.960i)6-s + 0.476·7-s + (−0.804 + 0.594i)8-s − 1.04·9-s + (−0.331 + 0.300i)10-s + 1.00·11-s + (−1.42 − 0.141i)12-s + 0.457·13-s + (0.319 + 0.353i)14-s − 0.640·15-s + (−0.980 − 0.197i)16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.969 - 0.246i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.969 - 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.254734 + 2.03842i\)
\(L(\frac12)\) \(\approx\) \(0.254734 + 2.03842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.949 - 1.04i)T \)
5 \( 1 - iT \)
23 \( 1 + (4.74 + 0.713i)T \)
good3 \( 1 - 2.47iT - 3T^{2} \)
7 \( 1 - 1.26T + 7T^{2} \)
11 \( 1 - 3.32T + 11T^{2} \)
13 \( 1 - 1.65T + 13T^{2} \)
17 \( 1 + 6.70iT - 17T^{2} \)
19 \( 1 + 0.390T + 19T^{2} \)
29 \( 1 - 8.11T + 29T^{2} \)
31 \( 1 + 2.86iT - 31T^{2} \)
37 \( 1 + 6.44iT - 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 2.71T + 43T^{2} \)
47 \( 1 - 0.827iT - 47T^{2} \)
53 \( 1 - 8.17iT - 53T^{2} \)
59 \( 1 - 4.19iT - 59T^{2} \)
61 \( 1 - 5.01iT - 61T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 - 9.80iT - 71T^{2} \)
73 \( 1 - 3.36T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 - 1.10T + 83T^{2} \)
89 \( 1 + 8.50iT - 89T^{2} \)
97 \( 1 - 7.78iT - 97T^{2} \)
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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

−11.51762046300091376529510066335, −10.62432016410925530609697558543, −9.527892195925020304112093939295, −8.872210088116512579196447448860, −7.75928606504816423893391500552, −6.67364159188318529685338839709, −5.67601455816756861865105053983, −4.60766616106242609368552561320, −3.99440469734202914129136381235, −2.84762487184491092627172563837, 1.20666264276522374159147871729, 1.94623946377542538174025954506, 3.60725988312691672988162563631, 4.74319404292150782182713211059, 6.15905600231019505009167315517, 6.51610941970317339312413179116, 8.014662048207496638068723516831, 8.673376197795191016703137430684, 9.938893633646080026423997466832, 10.96279781143286395477831190244

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