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Properties

Label 2-460-460.459-c1-0-52
Degree $2$
Conductor $460$
Sign $0.989 + 0.147i$
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  + 1.41·2-s + 1.41·3-s + 2.00·4-s − 2.23i·5-s + 2.00·6-s + 3.16i·7-s + 2.82·8-s − 0.999·9-s − 3.16i·10-s + 2.82·12-s + 4.47i·14-s − 3.16i·15-s + 4.00·16-s − 1.41·18-s − 4.47i·20-s + 4.47i·21-s + ⋯
L(s)  = 1  + 1.00·2-s + 0.816·3-s + 1.00·4-s − 0.999i·5-s + 0.816·6-s + 1.19i·7-s + 1.00·8-s − 0.333·9-s − 1.00i·10-s + 0.816·12-s + 1.19i·14-s − 0.816i·15-s + 1.00·16-s − 0.333·18-s − 1.00i·20-s + 0.975i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.989 + 0.147i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.989 + 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.15672 - 0.233995i\)
\(L(\frac12)\) \(\approx\) \(3.15672 - 0.233995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
5 \( 1 + 2.23iT \)
23 \( 1 + (-0.707 + 4.74i)T \)
good3 \( 1 - 1.41T + 3T^{2} \)
7 \( 1 - 3.16iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 - 3.16iT - 43T^{2} \)
47 \( 1 - 9.89T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 9.48iT - 83T^{2} \)
89 \( 1 + 17.8iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.46567928416907558943328408633, −10.11674083122167057227951229819, −8.915205570191278112913630210204, −8.508429071894576329032921565250, −7.39513620213291067077989308205, −5.97762438340083742079330086998, −5.33538228037219073388516601951, −4.20945450806335947052880341250, −2.96745284330156047879619934134, −1.96548294938586601885305370234, 2.05513735012747716929655619588, 3.34144313534881220043460161305, 3.82616216181584596204511580684, 5.31902583746644639849812701340, 6.49630381920731162955975914212, 7.34610902783289255079817421671, 7.952712527560932046263977753773, 9.435502936084605494759851098867, 10.45628614085111722957614490107, 11.07967004485684054454212851767

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