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Properties

Label 2-460-460.359-c1-0-10
Degree $2$
Conductor $460$
Sign $0.671 - 0.740i$
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  + (−1.35 + 0.398i)2-s + (0.305 + 0.352i)3-s + (1.68 − 1.08i)4-s + (−2.21 + 0.318i)5-s + (−0.554 − 0.356i)6-s + (4.70 + 2.14i)7-s + (−1.85 + 2.13i)8-s + (0.396 − 2.75i)9-s + (2.87 − 1.31i)10-s + (0.894 + 0.262i)12-s + (−7.24 − 1.04i)14-s + (−0.787 − 0.682i)15-s + (1.66 − 3.63i)16-s + (0.560 + 3.89i)18-s + (−3.37 + 2.92i)20-s + (0.679 + 2.31i)21-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (0.176 + 0.203i)3-s + (0.841 − 0.540i)4-s + (−0.989 + 0.142i)5-s + (−0.226 − 0.145i)6-s + (1.77 + 0.812i)7-s + (−0.654 + 0.755i)8-s + (0.132 − 0.918i)9-s + (0.909 − 0.415i)10-s + (0.258 + 0.0758i)12-s + (−1.93 − 0.278i)14-s + (−0.203 − 0.176i)15-s + (0.415 − 0.909i)16-s + (0.132 + 0.918i)18-s + (−0.755 + 0.654i)20-s + (0.148 + 0.504i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.671 - 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.920473 + 0.407862i\)
\(L(\frac12)\) \(\approx\) \(0.920473 + 0.407862i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.398i)T \)
5 \( 1 + (2.21 - 0.318i)T \)
23 \( 1 + (-0.657 - 4.75i)T \)
good3 \( 1 + (-0.305 - 0.352i)T + (-0.426 + 2.96i)T^{2} \)
7 \( 1 + (-4.70 - 2.14i)T + (4.58 + 5.29i)T^{2} \)
11 \( 1 + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-8.16 - 5.24i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (4.41 + 30.6i)T^{2} \)
37 \( 1 + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.130 - 0.907i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (3.18 - 2.76i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 7.98T + 47T^{2} \)
53 \( 1 + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (5.26 + 4.55i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (-3.82 - 13.0i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (-59.7 + 38.3i)T^{2} \)
73 \( 1 + (-30.3 + 66.4i)T^{2} \)
79 \( 1 + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (17.7 + 2.55i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-14.2 + 12.3i)T + (12.6 - 88.0i)T^{2} \)
97 \( 1 + (-93.0 + 27.3i)T^{2} \)
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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

−11.22391913552458614837791484921, −10.27535771216131610720856715088, −9.042520272128190247035680929253, −8.514140773602923837296479238407, −7.76466245003118523985697249106, −6.86545518755249932840018320168, −5.56875338677432678156813750369, −4.48903654520281567337789555446, −2.93913444815784302831596462349, −1.33040874748157163859120621651, 1.03979380110599341091132570570, 2.39639586542850070386312078498, 4.07092843755873378158411166466, 4.92904277750399554650144826203, 6.80592104937141110453434899824, 7.75022333180446098043857784314, 8.058872960382401180860190321279, 8.817021159938822911241700428492, 10.43055793226656471694965182285, 10.75537813011602715735671486404

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