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Properties

Label 2-464-29.12-c2-0-14
Degree $2$
Conductor $464$
Sign $0.625 - 0.780i$
Analytic cond. $12.6430$
Root an. cond. $3.55571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.11 + 3.11i)3-s − 4.53i·5-s + 0.745·7-s + 10.3i·9-s + (−0.0423 − 0.0423i)11-s + 8.30i·13-s + (14.0 − 14.0i)15-s + (15.2 + 15.2i)17-s + (25.1 + 25.1i)19-s + (2.31 + 2.31i)21-s − 8.64·23-s + 4.47·25-s + (−4.25 + 4.25i)27-s + (22.0 − 18.7i)29-s + (5.09 + 5.09i)31-s + ⋯
L(s)  = 1  + (1.03 + 1.03i)3-s − 0.906i·5-s + 0.106·7-s + 1.15i·9-s + (−0.00384 − 0.00384i)11-s + 0.638i·13-s + (0.939 − 0.939i)15-s + (0.897 + 0.897i)17-s + (1.32 + 1.32i)19-s + (0.110 + 0.110i)21-s − 0.375·23-s + 0.178·25-s + (−0.157 + 0.157i)27-s + (0.761 − 0.647i)29-s + (0.164 + 0.164i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.625 - 0.780i$
Analytic conductor: \(12.6430\)
Root analytic conductor: \(3.55571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1),\ 0.625 - 0.780i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.587200130\)
\(L(\frac12)\) \(\approx\) \(2.587200130\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-22.0 + 18.7i)T \)
good3 \( 1 + (-3.11 - 3.11i)T + 9iT^{2} \)
5 \( 1 + 4.53iT - 25T^{2} \)
7 \( 1 - 0.745T + 49T^{2} \)
11 \( 1 + (0.0423 + 0.0423i)T + 121iT^{2} \)
13 \( 1 - 8.30iT - 169T^{2} \)
17 \( 1 + (-15.2 - 15.2i)T + 289iT^{2} \)
19 \( 1 + (-25.1 - 25.1i)T + 361iT^{2} \)
23 \( 1 + 8.64T + 529T^{2} \)
31 \( 1 + (-5.09 - 5.09i)T + 961iT^{2} \)
37 \( 1 + (-10.0 + 10.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (-2.50 + 2.50i)T - 1.68e3iT^{2} \)
43 \( 1 + (-11.2 - 11.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-23.0 + 23.0i)T - 2.20e3iT^{2} \)
53 \( 1 + 42.8T + 2.80e3T^{2} \)
59 \( 1 + 106.T + 3.48e3T^{2} \)
61 \( 1 + (42.3 + 42.3i)T + 3.72e3iT^{2} \)
67 \( 1 + 75.7iT - 4.48e3T^{2} \)
71 \( 1 + 71.2iT - 5.04e3T^{2} \)
73 \( 1 + (73.7 - 73.7i)T - 5.32e3iT^{2} \)
79 \( 1 + (-78.7 - 78.7i)T + 6.24e3iT^{2} \)
83 \( 1 + 15.1T + 6.88e3T^{2} \)
89 \( 1 + (22.8 + 22.8i)T + 7.92e3iT^{2} \)
97 \( 1 + (-42.8 + 42.8i)T - 9.40e3iT^{2} \)
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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

−10.67448640679045861653017978782, −9.785100908681917741037123678004, −9.304917499812775072153746313341, −8.301136353589078853795203613392, −7.80399286106071513260568234316, −6.09935362146492079878747273579, −4.95937605624715149423023805819, −4.07441614061657918888186984362, −3.15732888606475692034809368439, −1.50902207644582822788117651922, 1.12062734956763570741676631182, 2.77851515113990855817547835520, 3.11125329880456511734890126257, 4.98537872809410524177200882137, 6.34053288455974651499066351842, 7.34849871077447631902720864440, 7.65799599196094910052530530877, 8.799454572005257985643166171268, 9.683152336359464310440153103528, 10.71449993171976433671315267479

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