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Properties

Label 2-460-92.91-c1-0-33
Degree $2$
Conductor $460$
Sign $-0.897 + 0.441i$
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  + (−0.637 − 1.26i)2-s + 2.87i·3-s + (−1.18 + 1.61i)4-s − i·5-s + (3.62 − 1.83i)6-s − 2.68·7-s + (2.78 + 0.469i)8-s − 5.25·9-s + (−1.26 + 0.637i)10-s − 4.89·11-s + (−4.62 − 3.40i)12-s + 0.727·13-s + (1.71 + 3.38i)14-s + 2.87·15-s + (−1.18 − 3.82i)16-s − 6.34i·17-s + ⋯
L(s)  = 1  + (−0.451 − 0.892i)2-s + 1.65i·3-s + (−0.593 + 0.805i)4-s − 0.447i·5-s + (1.48 − 0.748i)6-s − 1.01·7-s + (0.986 + 0.166i)8-s − 1.75·9-s + (−0.399 + 0.201i)10-s − 1.47·11-s + (−1.33 − 0.983i)12-s + 0.201·13-s + (0.457 + 0.904i)14-s + 0.741·15-s + (−0.296 − 0.955i)16-s − 1.53i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.897 + 0.441i$
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: \((2,\ 460,\ (\ :1/2),\ -0.897 + 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0182293 - 0.0783734i\)
\(L(\frac12)\) \(\approx\) \(0.0182293 - 0.0783734i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.637 + 1.26i)T \)
5 \( 1 + iT \)
23 \( 1 + (-2.20 + 4.25i)T \)
good3 \( 1 - 2.87iT - 3T^{2} \)
7 \( 1 + 2.68T + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 0.727T + 13T^{2} \)
17 \( 1 + 6.34iT - 17T^{2} \)
19 \( 1 - 2.68T + 19T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 - 6.13iT - 31T^{2} \)
37 \( 1 + 3.87iT - 37T^{2} \)
41 \( 1 + 8.07T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 9.80iT - 47T^{2} \)
53 \( 1 - 11.1iT - 53T^{2} \)
59 \( 1 + 1.87iT - 59T^{2} \)
61 \( 1 - 6.93iT - 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 - 7.24iT - 71T^{2} \)
73 \( 1 - 7.90T + 73T^{2} \)
79 \( 1 + 3.67T + 79T^{2} \)
83 \( 1 - 4.75T + 83T^{2} \)
89 \( 1 + 8.12iT - 89T^{2} \)
97 \( 1 + 2.68iT - 97T^{2} \)
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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

−10.37639513466634669077534260991, −9.968759733441267605709911281800, −9.166605155930083334203777905207, −8.517905015283230929283246203452, −7.21021288149269082380065576749, −5.33083195941804661984752884599, −4.78132255548904087076555883419, −3.47900116940985218018202864017, −2.79589200665426229532317916033, −0.05531188581005735046006962523, 1.75842431750839318442323475547, 3.28652383827577006801886914786, 5.37534133821970313371183255947, 6.21157318663622668423845138266, 6.84252379886892267954377924890, 7.78906640387482755890890860154, 8.198780516796029493967779590314, 9.523931766894998811913125613125, 10.40291109561354958678510307835, 11.41598326762085838329689899691

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