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Properties

Label 2-460-92.91-c1-0-25
Degree $2$
Conductor $460$
Sign $0.692 + 0.721i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.692 + 0.721i$
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: \((2,\ 460,\ (\ :1/2),\ 0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05020 - 0.447574i\)
\(L(\frac12)\) \(\approx\) \(1.05020 - 0.447574i\)
\(L(\frac{3}{2})\) 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

−11.05406254707587609986835408916, −10.06275096172492849263707679510, −8.665369656950444230613155749038, −8.305609312406728307873500893412, −7.55170123642167652671838030114, −6.42847249025707788315825650011, −5.61999675400231677824812957722, −4.15454458246071619557033019736, −1.98872000927601495813380200073, −1.22206993851212959884778812761, 1.47852036871197510685108011312, 3.03154882026283112225820014468, 4.23224827309816665260628431922, 5.41002059449978801325032495293, 6.85422090600121374224275130107, 7.68125969054116938871640222765, 8.706883487261174996457718164273, 9.318686623792450705994292495612, 10.46545036594870110112514114468, 11.01674353849628530223902870341

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