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Properties

Label 2-460-92.91-c1-0-32
Degree $2$
Conductor $460$
Sign $0.999 - 0.0302i$
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  + (1.15 + 0.812i)2-s − 1.55i·3-s + (0.681 + 1.88i)4-s − i·5-s + (1.26 − 1.80i)6-s + 2.00·7-s + (−0.738 + 2.73i)8-s + 0.575·9-s + (0.812 − 1.15i)10-s − 4.45·11-s + (2.92 − 1.06i)12-s + 6.20·13-s + (2.31 + 1.62i)14-s − 1.55·15-s + (−3.07 + 2.56i)16-s − 3.16i·17-s + ⋯
L(s)  = 1  + (0.818 + 0.574i)2-s − 0.898i·3-s + (0.340 + 0.940i)4-s − 0.447i·5-s + (0.516 − 0.736i)6-s + 0.757·7-s + (−0.261 + 0.965i)8-s + 0.191·9-s + (0.256 − 0.366i)10-s − 1.34·11-s + (0.845 − 0.306i)12-s + 1.72·13-s + (0.619 + 0.434i)14-s − 0.402·15-s + (−0.767 + 0.640i)16-s − 0.767i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.999 - 0.0302i$
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: \((2,\ 460,\ (\ :1/2),\ 0.999 - 0.0302i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39592 + 0.0362517i\)
\(L(\frac12)\) \(\approx\) \(2.39592 + 0.0362517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.812i)T \)
5 \( 1 + iT \)
23 \( 1 + (4.55 + 1.49i)T \)
good3 \( 1 + 1.55iT - 3T^{2} \)
7 \( 1 - 2.00T + 7T^{2} \)
11 \( 1 + 4.45T + 11T^{2} \)
13 \( 1 - 6.20T + 13T^{2} \)
17 \( 1 + 3.16iT - 17T^{2} \)
19 \( 1 - 6.11T + 19T^{2} \)
29 \( 1 + 2.57T + 29T^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 - 8.10iT - 37T^{2} \)
41 \( 1 + 8.07T + 41T^{2} \)
43 \( 1 - 2.38T + 43T^{2} \)
47 \( 1 - 1.24iT - 47T^{2} \)
53 \( 1 - 12.6iT - 53T^{2} \)
59 \( 1 - 8.78iT - 59T^{2} \)
61 \( 1 + 7.69iT - 61T^{2} \)
67 \( 1 + 6.98T + 67T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 + 2.40T + 79T^{2} \)
83 \( 1 - 0.858T + 83T^{2} \)
89 \( 1 + 5.91iT - 89T^{2} \)
97 \( 1 - 5.79iT - 97T^{2} \)
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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

−11.47415753094983496085974307349, −10.36138901290229135140310257504, −8.821998577964022107230972485543, −7.929708642705436609037457097472, −7.51665316785270594224039176417, −6.31567351326704203679851882655, −5.41866217742635833598196503138, −4.50200786366114958164130548868, −3.07596337318972180007689097621, −1.54728576581977107218055240458, 1.74388990068865505226136818889, 3.32852568281342851243004282521, 4.06211143615645508754276857070, 5.22858719980822617159887304668, 5.89360000937322481123460080044, 7.32130521849825638173935959430, 8.414051898922006963211453616571, 9.713256156779701495809113739121, 10.39181675006743675390076736585, 11.02948711881048014278667709272

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