Properties

Label 2-464-1.1-c3-0-22
Degree $2$
Conductor $464$
Sign $1$
Analytic cond. $27.3768$
Root an. cond. $5.23229$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.24·3-s + 0.656·5-s − 6.14·7-s + 58.4·9-s + 65.3·11-s − 49.7·13-s + 6.07·15-s + 55.4·17-s + 64.7·19-s − 56.7·21-s − 93.8·23-s − 124.·25-s + 290.·27-s + 29·29-s + 236.·31-s + 603.·33-s − 4.03·35-s + 76.8·37-s − 460.·39-s + 215.·41-s − 80.8·43-s + 38.3·45-s + 357.·47-s − 305.·49-s + 512.·51-s + 328.·53-s + 42.9·55-s + ⋯
L(s)  = 1  + 1.77·3-s + 0.0587·5-s − 0.331·7-s + 2.16·9-s + 1.79·11-s − 1.06·13-s + 0.104·15-s + 0.791·17-s + 0.781·19-s − 0.589·21-s − 0.851·23-s − 0.996·25-s + 2.07·27-s + 0.185·29-s + 1.36·31-s + 3.18·33-s − 0.0194·35-s + 0.341·37-s − 1.88·39-s + 0.819·41-s − 0.286·43-s + 0.127·45-s + 1.11·47-s − 0.890·49-s + 1.40·51-s + 0.851·53-s + 0.105·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $1$
Analytic conductor: \(27.3768\)
Root analytic conductor: \(5.23229\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.988407424\)
\(L(\frac12)\) \(\approx\) \(3.988407424\)
\(L(\frac{5}{2})\) 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 \( 1 - 29T \)
good3 \( 1 - 9.24T + 27T^{2} \)
5 \( 1 - 0.656T + 125T^{2} \)
7 \( 1 + 6.14T + 343T^{2} \)
11 \( 1 - 65.3T + 1.33e3T^{2} \)
13 \( 1 + 49.7T + 2.19e3T^{2} \)
17 \( 1 - 55.4T + 4.91e3T^{2} \)
19 \( 1 - 64.7T + 6.85e3T^{2} \)
23 \( 1 + 93.8T + 1.21e4T^{2} \)
31 \( 1 - 236.T + 2.97e4T^{2} \)
37 \( 1 - 76.8T + 5.06e4T^{2} \)
41 \( 1 - 215.T + 6.89e4T^{2} \)
43 \( 1 + 80.8T + 7.95e4T^{2} \)
47 \( 1 - 357.T + 1.03e5T^{2} \)
53 \( 1 - 328.T + 1.48e5T^{2} \)
59 \( 1 - 99.2T + 2.05e5T^{2} \)
61 \( 1 + 725.T + 2.26e5T^{2} \)
67 \( 1 + 844.T + 3.00e5T^{2} \)
71 \( 1 - 378.T + 3.57e5T^{2} \)
73 \( 1 + 581.T + 3.89e5T^{2} \)
79 \( 1 - 353.T + 4.93e5T^{2} \)
83 \( 1 + 696.T + 5.71e5T^{2} \)
89 \( 1 - 1.11e3T + 7.04e5T^{2} \)
97 \( 1 + 805.T + 9.12e5T^{2} \)
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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

−10.07555892973262500665317721605, −9.617928620828343113349283360001, −8.954385336525583748073464215368, −7.918083194474415170445556792178, −7.25999113088945968858502194409, −6.13289769561649529171874232601, −4.41824193807745198649569365538, −3.58581447039045162228379826295, −2.57973953734593212920978969947, −1.34340706674633016349526612836, 1.34340706674633016349526612836, 2.57973953734593212920978969947, 3.58581447039045162228379826295, 4.41824193807745198649569365538, 6.13289769561649529171874232601, 7.25999113088945968858502194409, 7.918083194474415170445556792178, 8.954385336525583748073464215368, 9.617928620828343113349283360001, 10.07555892973262500665317721605

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