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Properties

Label 2-460-92.91-c1-0-1
Degree $2$
Conductor $460$
Sign $-0.595 + 0.803i$
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  + (−1.05 + 0.945i)2-s + 1.57i·3-s + (0.210 − 1.98i)4-s − i·5-s + (−1.49 − 1.65i)6-s − 1.14·7-s + (1.66 + 2.28i)8-s + 0.509·9-s + (0.945 + 1.05i)10-s − 5.86·11-s + (3.13 + 0.331i)12-s − 2.69·13-s + (1.20 − 1.08i)14-s + 1.57·15-s + (−3.91 − 0.836i)16-s − 0.337i·17-s + ⋯
L(s)  = 1  + (−0.743 + 0.668i)2-s + 0.911i·3-s + (0.105 − 0.994i)4-s − 0.447i·5-s + (−0.609 − 0.677i)6-s − 0.434·7-s + (0.587 + 0.809i)8-s + 0.169·9-s + (0.299 + 0.332i)10-s − 1.76·11-s + (0.906 + 0.0958i)12-s − 0.747·13-s + (0.322 − 0.290i)14-s + 0.407·15-s + (−0.977 − 0.209i)16-s − 0.0818i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.595 + 0.803i$
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: \((2,\ 460,\ (\ :1/2),\ -0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0451473 - 0.0896920i\)
\(L(\frac12)\) \(\approx\) \(0.0451473 - 0.0896920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.05 - 0.945i)T \)
5 \( 1 + iT \)
23 \( 1 + (2.43 - 4.13i)T \)
good3 \( 1 - 1.57iT - 3T^{2} \)
7 \( 1 + 1.14T + 7T^{2} \)
11 \( 1 + 5.86T + 11T^{2} \)
13 \( 1 + 2.69T + 13T^{2} \)
17 \( 1 + 0.337iT - 17T^{2} \)
19 \( 1 + 4.49T + 19T^{2} \)
29 \( 1 + 9.68T + 29T^{2} \)
31 \( 1 + 4.80iT - 31T^{2} \)
37 \( 1 + 3.78iT - 37T^{2} \)
41 \( 1 - 7.24T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + 2.80iT - 47T^{2} \)
53 \( 1 - 7.11iT - 53T^{2} \)
59 \( 1 - 1.70iT - 59T^{2} \)
61 \( 1 + 13.9iT - 61T^{2} \)
67 \( 1 + 6.00T + 67T^{2} \)
71 \( 1 - 3.87iT - 71T^{2} \)
73 \( 1 + 1.87T + 73T^{2} \)
79 \( 1 - 5.07T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 - 18.1iT - 89T^{2} \)
97 \( 1 - 14.5iT - 97T^{2} \)
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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

−11.10289945099147378760013804102, −10.52047749159189263556283312050, −9.636100261845870251761036582086, −9.226444618997224951147070792394, −7.909073184872350884824858662561, −7.38668129487851204611818189341, −5.88312298538039930883994289382, −5.16482345257369886617953990816, −4.12545607960277889003133240219, −2.29309415791752375714191238498, 0.07212257772554725741781121816, 2.05344407151291209146616209593, 2.85643932348975605392109631495, 4.39861932606605039853307328065, 6.03048620002833340044838729427, 7.19617469816492974272095206443, 7.64205722408345852679696411039, 8.586480361268009442623555342073, 9.827108420895073642842278489907, 10.43362143224445802931057639578

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