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Properties

Label 2-460-460.79-c1-0-8
Degree $2$
Conductor $460$
Sign $0.433 + 0.901i$
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  + (−0.926 − 1.06i)2-s + (−2.78 − 1.79i)3-s + (−0.284 + 1.97i)4-s + (−2.03 − 0.928i)5-s + (0.667 + 4.64i)6-s + (−0.518 + 1.76i)7-s + (2.37 − 1.52i)8-s + (3.32 + 7.27i)9-s + (0.890 + 3.03i)10-s + (4.34 − 5.01i)12-s + (2.36 − 1.08i)14-s + (4.00 + 6.23i)15-s + (−3.83 − 1.12i)16-s + (4.69 − 10.2i)18-s + (2.41 − 3.76i)20-s + (4.60 − 3.99i)21-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−1.61 − 1.03i)3-s + (−0.142 + 0.989i)4-s + (−0.909 − 0.415i)5-s + (0.272 + 1.89i)6-s + (−0.195 + 0.666i)7-s + (0.841 − 0.540i)8-s + (1.10 + 2.42i)9-s + (0.281 + 0.959i)10-s + (1.25 − 1.44i)12-s + (0.632 − 0.288i)14-s + (1.03 + 1.61i)15-s + (−0.959 − 0.281i)16-s + (1.10 − 2.42i)18-s + (0.540 − 0.841i)20-s + (1.00 − 0.871i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.433 + 0.901i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.433 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307395 - 0.193274i\)
\(L(\frac12)\) \(\approx\) \(0.307395 - 0.193274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.926 + 1.06i)T \)
5 \( 1 + (2.03 + 0.928i)T \)
23 \( 1 + (4.04 - 2.57i)T \)
good3 \( 1 + (2.78 + 1.79i)T + (1.24 + 2.72i)T^{2} \)
7 \( 1 + (0.518 - 1.76i)T + (-5.88 - 3.78i)T^{2} \)
11 \( 1 + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.803 + 5.58i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-12.8 + 28.1i)T^{2} \)
37 \( 1 + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-5.30 + 11.6i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (6.56 - 10.2i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-3.24 - 5.05i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-11.7 + 10.2i)T + (9.53 - 66.3i)T^{2} \)
71 \( 1 + (10.1 - 70.2i)T^{2} \)
73 \( 1 + (70.0 + 20.5i)T^{2} \)
79 \( 1 + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-14.9 + 6.80i)T + (54.3 - 62.7i)T^{2} \)
89 \( 1 + (3.88 - 6.04i)T + (-36.9 - 80.9i)T^{2} \)
97 \( 1 + (-63.5 - 73.3i)T^{2} \)
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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

−11.17331426988500871937709513983, −10.33853348229515344719695109715, −9.137553205931128370932847295712, −7.968499753169123827899960243560, −7.43796469769065273373025906102, −6.30511738718551853932947062646, −5.22999022903905044717471820565, −4.04798839339012823184679458988, −2.18169066644262755315997538412, −0.72859393863334251593658608276, 0.59488288583165618830603866263, 3.81879832071668360288749407145, 4.62968154830117277973939891182, 5.66257696473119376186459913088, 6.63229031542145662353279294474, 7.25022808603019935861246456710, 8.529003495933725332908292453944, 9.706532859754653888391061391071, 10.39385074386226611500933004693, 10.93251801810448806883913594878

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