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Properties

Label 2-460-460.359-c1-0-62
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

\(L(1)\) \(\approx\) \(0.569915 - 1.28619i\)
\(L(\frac12)\) \(\approx\) \(0.569915 - 1.28619i\)
\(L(\frac{3}{2})\) 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

−10.76722593513507633179557779058, −10.17447196416405123510491340555, −9.113483404213237965883761663668, −7.58598719747545193306391458045, −6.59523388803545744198995507786, −6.39921445614572809540924862280, −4.65664535344883287707288051961, −3.64996705926041600140151295478, −3.05222131478632543963866758981, −0.63878598747482273801789257506, 2.62707090545114813140643196530, 3.56742211413299554026538781173, 4.64009927265258481126138405851, 5.71067131357341260913337466184, 6.58179458698345710524202597439, 7.55727833908934960666351366867, 8.477917128492783448894663336011, 9.689008669223046492194912037768, 10.67065974529006477473243341033, 11.73121793579125157331187572156

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