Properties

Label 2-460-92.91-c1-0-23
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 − 1.47i·3-s + (1.68 − 1.07i)4-s + i·5-s + (0.583 + 1.99i)6-s − 3.53·7-s + (−1.86 + 2.12i)8-s + 0.833·9-s + (−0.396 − 1.35i)10-s + 1.16·11-s + (−1.58 − 2.48i)12-s + 5.75·13-s + (4.80 − 1.40i)14-s + 1.47·15-s + (1.68 − 3.62i)16-s − 6.92i·17-s + ⋯
L(s)  = 1  + (−0.959 + 0.280i)2-s − 0.849i·3-s + (0.843 − 0.537i)4-s + 0.447i·5-s + (0.238 + 0.815i)6-s − 1.33·7-s + (−0.658 + 0.752i)8-s + 0.277·9-s + (−0.125 − 0.429i)10-s + 0.350·11-s + (−0.456 − 0.716i)12-s + 1.59·13-s + (1.28 − 0.374i)14-s + 0.380·15-s + (0.421 − 0.906i)16-s − 1.67i·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

\(L(1)\) \(\approx\) \(0.676011 - 0.462582i\)
\(L(\frac12)\) \(\approx\) \(0.676011 - 0.462582i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.47iT - 3T^{2} \)
7 \( 1 + 3.53T + 7T^{2} \)
11 \( 1 - 1.16T + 11T^{2} \)
13 \( 1 - 5.75T + 13T^{2} \)
17 \( 1 + 6.92iT - 17T^{2} \)
19 \( 1 - 3.53T + 19T^{2} \)
29 \( 1 + 5.71T + 29T^{2} \)
31 \( 1 + 8.33iT - 31T^{2} \)
37 \( 1 + 2.93iT - 37T^{2} \)
41 \( 1 - 6.63T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 + 0.327iT - 47T^{2} \)
53 \( 1 - 7.45iT - 53T^{2} \)
59 \( 1 + 8.51iT - 59T^{2} \)
61 \( 1 + 8.40iT - 61T^{2} \)
67 \( 1 + 3.74T + 67T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 - 1.85T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 - 14.9iT - 89T^{2} \)
97 \( 1 + 9.46iT - 97T^{2} \)
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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

−10.85647641417368630061677526476, −9.589532512795753682059450300622, −9.382494654723425970623788767041, −7.936058691909872912165487151467, −7.24594803136078960831890122007, −6.43706829150304286679104319117, −5.84840570495064237173181239473, −3.66510400874637981374377610599, −2.36484423161520116060606122481, −0.76397020552538127452632406971, 1.43954507740041303317078952382, 3.45663465378378380230523285508, 3.91948233866722059370875747032, 5.83433375695896183650306891002, 6.57343380090334543730323573953, 7.85199858359680635724467179181, 8.916404884135814113597286923133, 9.379314889775270804899193019944, 10.31474114441603328142460919423, 10.77690701375818549789387419533

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