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Properties

Label 2-460-92.91-c1-0-22
Degree $2$
Conductor $460$
Sign $0.362 - 0.932i$
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 + 0.396i)2-s + 0.679i·3-s + (1.68 + 1.07i)4-s + i·5-s + (−0.269 + 0.922i)6-s + 1.16·7-s + (1.86 + 2.12i)8-s + 2.53·9-s + (−0.396 + 1.35i)10-s − 3.53·11-s + (−0.730 + 1.14i)12-s − 2.38·13-s + (1.57 + 0.460i)14-s − 0.679·15-s + (1.68 + 3.62i)16-s − 3.19i·17-s + ⋯
L(s)  = 1  + (0.959 + 0.280i)2-s + 0.392i·3-s + (0.843 + 0.537i)4-s + 0.447i·5-s + (−0.109 + 0.376i)6-s + 0.439·7-s + (0.658 + 0.752i)8-s + 0.846·9-s + (−0.125 + 0.429i)10-s − 1.06·11-s + (−0.210 + 0.330i)12-s − 0.661·13-s + (0.421 + 0.123i)14-s − 0.175·15-s + (0.421 + 0.906i)16-s − 0.775i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.362 - 0.932i$
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: \((2,\ 460,\ (\ :1/2),\ 0.362 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13140 + 1.45848i\)
\(L(\frac12)\) \(\approx\) \(2.13140 + 1.45848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.396i)T \)
5 \( 1 - iT \)
23 \( 1 + (-4.70 + 0.939i)T \)
good3 \( 1 - 0.679iT - 3T^{2} \)
7 \( 1 - 1.16T + 7T^{2} \)
11 \( 1 + 3.53T + 11T^{2} \)
13 \( 1 + 2.38T + 13T^{2} \)
17 \( 1 + 3.19iT - 17T^{2} \)
19 \( 1 + 1.16T + 19T^{2} \)
29 \( 1 + 0.284T + 29T^{2} \)
31 \( 1 + 4.43iT - 31T^{2} \)
37 \( 1 - 8.93iT - 37T^{2} \)
41 \( 1 + 2.51T + 41T^{2} \)
43 \( 1 - 6.07T + 43T^{2} \)
47 \( 1 + 13.2iT - 47T^{2} \)
53 \( 1 + 7.45iT - 53T^{2} \)
59 \( 1 + 8.51iT - 59T^{2} \)
61 \( 1 + 7.71iT - 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + 1.46iT - 71T^{2} \)
73 \( 1 - 4.88T + 73T^{2} \)
79 \( 1 + 5.08T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 - 3.06iT - 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−11.25394776786527706079508197933, −10.51947823230910787862025364887, −9.663200876922699703007477596945, −8.195372873881755094294624559901, −7.37407122140345119845452983096, −6.61392337559062159730564319816, −5.18874877208672898391856343274, −4.70413416733843818220883361948, −3.38474904921978129417969595936, −2.23552376405808972608003418169, 1.43856864676911672686469167644, 2.68621078566730402084723493977, 4.21383679475560549731530733427, 4.99991443071908966721085700919, 5.98118516620320730586311746570, 7.21779622089498157433371716322, 7.80684249526518122496978714895, 9.188143279699344956040421241402, 10.36523874192750208997447487004, 10.88677701407664135066660327803

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