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Properties

Label 2-464-29.12-c2-0-12
Degree $2$
Conductor $464$
Sign $0.904 - 0.425i$
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  + (−0.442 − 0.442i)3-s + 4.16i·5-s + 9.68·7-s − 8.60i·9-s + (0.334 + 0.334i)11-s + 12.2i·13-s + (1.84 − 1.84i)15-s + (6.80 + 6.80i)17-s + (−14.6 − 14.6i)19-s + (−4.28 − 4.28i)21-s + 10.0·23-s + 7.65·25-s + (−7.79 + 7.79i)27-s + (28.1 − 7.15i)29-s + (37.3 + 37.3i)31-s + ⋯
L(s)  = 1  + (−0.147 − 0.147i)3-s + 0.832i·5-s + 1.38·7-s − 0.956i·9-s + (0.0304 + 0.0304i)11-s + 0.940i·13-s + (0.122 − 0.122i)15-s + (0.400 + 0.400i)17-s + (−0.771 − 0.771i)19-s + (−0.204 − 0.204i)21-s + 0.438·23-s + 0.306·25-s + (−0.288 + 0.288i)27-s + (0.969 − 0.246i)29-s + (1.20 + 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.904 - 0.425i$
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: \((2,\ 464,\ (\ :1),\ 0.904 - 0.425i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.942215016\)
\(L(\frac12)\) \(\approx\) \(1.942215016\)
\(L(2)\) 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 \( 1 + (-28.1 + 7.15i)T \)
good3 \( 1 + (0.442 + 0.442i)T + 9iT^{2} \)
5 \( 1 - 4.16iT - 25T^{2} \)
7 \( 1 - 9.68T + 49T^{2} \)
11 \( 1 + (-0.334 - 0.334i)T + 121iT^{2} \)
13 \( 1 - 12.2iT - 169T^{2} \)
17 \( 1 + (-6.80 - 6.80i)T + 289iT^{2} \)
19 \( 1 + (14.6 + 14.6i)T + 361iT^{2} \)
23 \( 1 - 10.0T + 529T^{2} \)
31 \( 1 + (-37.3 - 37.3i)T + 961iT^{2} \)
37 \( 1 + (45.0 - 45.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (-22.8 + 22.8i)T - 1.68e3iT^{2} \)
43 \( 1 + (-17.5 - 17.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (2.20 - 2.20i)T - 2.20e3iT^{2} \)
53 \( 1 - 90.1T + 2.80e3T^{2} \)
59 \( 1 - 90.4T + 3.48e3T^{2} \)
61 \( 1 + (29.4 + 29.4i)T + 3.72e3iT^{2} \)
67 \( 1 - 31.5iT - 4.48e3T^{2} \)
71 \( 1 + 99.8iT - 5.04e3T^{2} \)
73 \( 1 + (3.96 - 3.96i)T - 5.32e3iT^{2} \)
79 \( 1 + (-40.3 - 40.3i)T + 6.24e3iT^{2} \)
83 \( 1 + 137.T + 6.88e3T^{2} \)
89 \( 1 + (-83.1 - 83.1i)T + 7.92e3iT^{2} \)
97 \( 1 + (79.9 - 79.9i)T - 9.40e3iT^{2} \)
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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

−10.94138828386127859253965440314, −10.21401252728583374009697374475, −8.945755122136457898883310332130, −8.265099973885001879486634115024, −6.95482990529396112476702298919, −6.53325404031763786123286599747, −5.10997784279677488800392436138, −4.10689016213240526146152584375, −2.71738061548442059086808685473, −1.27703759903256175971625658677, 1.01950833652672568761404715083, 2.39262485129313631344768199890, 4.21814990122821352764188306416, 5.04377534627846295257820094721, 5.69829029450058677875323143983, 7.35102710220696327804009607448, 8.215458066386088179763412089283, 8.653248443098351437987702056559, 10.08759918006154901148163166730, 10.73303651356948801049001484463

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