Properties

Label 2-460-115.104-c1-0-3
Degree $2$
Conductor $460$
Sign $0.952 - 0.303i$
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  + (−2.25 − 0.324i)3-s + (−0.690 − 2.12i)5-s + (−3.55 + 3.08i)7-s + (2.10 + 0.618i)9-s + (1.12 + 0.725i)11-s + (3.67 + 3.18i)13-s + (0.867 + 5.02i)15-s + (5.55 − 2.53i)17-s + (1.84 − 4.04i)19-s + (9.02 − 5.80i)21-s + (−0.167 − 4.79i)23-s + (−4.04 + 2.93i)25-s + (1.67 + 0.763i)27-s + (1.39 + 3.04i)29-s + (1.02 + 7.11i)31-s + ⋯
L(s)  = 1  + (−1.30 − 0.187i)3-s + (−0.308 − 0.951i)5-s + (−1.34 + 1.16i)7-s + (0.701 + 0.206i)9-s + (0.340 + 0.218i)11-s + (1.01 + 0.882i)13-s + (0.224 + 1.29i)15-s + (1.34 − 0.615i)17-s + (0.423 − 0.928i)19-s + (1.97 − 1.26i)21-s + (−0.0349 − 0.999i)23-s + (−0.809 + 0.587i)25-s + (0.321 + 0.146i)27-s + (0.258 + 0.565i)29-s + (0.183 + 1.27i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.952 - 0.303i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.952 - 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.696326 + 0.108238i\)
\(L(\frac12)\) \(\approx\) \(0.696326 + 0.108238i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.690 + 2.12i)T \)
23 \( 1 + (0.167 + 4.79i)T \)
good3 \( 1 + (2.25 + 0.324i)T + (2.87 + 0.845i)T^{2} \)
7 \( 1 + (3.55 - 3.08i)T + (0.996 - 6.92i)T^{2} \)
11 \( 1 + (-1.12 - 0.725i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-3.67 - 3.18i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (-5.55 + 2.53i)T + (11.1 - 12.8i)T^{2} \)
19 \( 1 + (-1.84 + 4.04i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-1.39 - 3.04i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-1.02 - 7.11i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (0.942 - 3.20i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (-1.71 + 0.502i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-5.67 - 0.815i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 - 11.3iT - 47T^{2} \)
53 \( 1 + (-5.72 + 4.95i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (2.17 - 2.50i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (-1.79 - 12.4i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-2.75 - 4.28i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (1.25 - 0.804i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (12.6 + 5.76i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (-3.76 + 4.34i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (0.233 - 0.795i)T + (-69.8 - 44.8i)T^{2} \)
89 \( 1 + (-0.786 + 5.47i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-2.67 - 9.11i)T + (-81.6 + 52.4i)T^{2} \)
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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

−11.44191432991028631406590422340, −10.23472113833362460020953484131, −9.168742241484646011575292122199, −8.725892744578585259504834624272, −7.10980667142936467454176683225, −6.25022967218839722110196058154, −5.56263766615892012443939174844, −4.59607109638361631632319419588, −3.09366889939574693658705208047, −1.00645342480032333237144964223, 0.70786466789567605587966598366, 3.39914718953464391604968555904, 3.87745013963169847804930550861, 5.82921340009999407978004511193, 6.05591948225698455247924063218, 7.16829063022379052730859539962, 7.987497219537105952120230525811, 9.782368738947910265615815382918, 10.26214386722433544290768932685, 10.90255864646785295169338172427

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