Properties

Label 2-507-1.1-c3-0-27
Degree $2$
Conductor $507$
Sign $1$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
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  − 1.46·2-s − 3·3-s − 5.85·4-s + 9.85·5-s + 4.39·6-s + 29.9·7-s + 20.3·8-s + 9·9-s − 14.4·10-s + 46.9·11-s + 17.5·12-s − 43.8·14-s − 29.5·15-s + 17.0·16-s − 48.2·17-s − 13.1·18-s + 120.·19-s − 57.6·20-s − 89.8·21-s − 68.7·22-s + 130.·23-s − 60.9·24-s − 27.9·25-s − 27·27-s − 175.·28-s − 194.·29-s + 43.3·30-s + ⋯
L(s)  = 1  − 0.518·2-s − 0.577·3-s − 0.731·4-s + 0.881·5-s + 0.299·6-s + 1.61·7-s + 0.897·8-s + 0.333·9-s − 0.456·10-s + 1.28·11-s + 0.422·12-s − 0.837·14-s − 0.508·15-s + 0.266·16-s − 0.688·17-s − 0.172·18-s + 1.45·19-s − 0.644·20-s − 0.933·21-s − 0.666·22-s + 1.18·23-s − 0.518·24-s − 0.223·25-s − 0.192·27-s − 1.18·28-s − 1.24·29-s + 0.263·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.671041340\)
\(L(\frac12)\) \(\approx\) \(1.671041340\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 \)
good2 \( 1 + 1.46T + 8T^{2} \)
5 \( 1 - 9.85T + 125T^{2} \)
7 \( 1 - 29.9T + 343T^{2} \)
11 \( 1 - 46.9T + 1.33e3T^{2} \)
17 \( 1 + 48.2T + 4.91e3T^{2} \)
19 \( 1 - 120.T + 6.85e3T^{2} \)
23 \( 1 - 130.T + 1.21e4T^{2} \)
29 \( 1 + 194.T + 2.43e4T^{2} \)
31 \( 1 + 32.0T + 2.97e4T^{2} \)
37 \( 1 + 32.4T + 5.06e4T^{2} \)
41 \( 1 + 241.T + 6.89e4T^{2} \)
43 \( 1 - 96.4T + 7.95e4T^{2} \)
47 \( 1 - 539.T + 1.03e5T^{2} \)
53 \( 1 + 152.T + 1.48e5T^{2} \)
59 \( 1 - 327.T + 2.05e5T^{2} \)
61 \( 1 + 98.4T + 2.26e5T^{2} \)
67 \( 1 + 441.T + 3.00e5T^{2} \)
71 \( 1 - 345.T + 3.57e5T^{2} \)
73 \( 1 - 773.T + 3.89e5T^{2} \)
79 \( 1 + 150.T + 4.93e5T^{2} \)
83 \( 1 - 337.T + 5.71e5T^{2} \)
89 \( 1 - 169.T + 7.04e5T^{2} \)
97 \( 1 - 214.T + 9.12e5T^{2} \)
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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

−10.50795284610599784451485811020, −9.368368686525530344826145041238, −9.035718570315973199417007272758, −7.86063465600745783264813086438, −6.97092242988088816256993614523, −5.59768319690431338580334897316, −4.98461996424065223444609487328, −3.95632270439465503362367682100, −1.79219927169727868401328167695, −1.01049147697519497454147865268, 1.01049147697519497454147865268, 1.79219927169727868401328167695, 3.95632270439465503362367682100, 4.98461996424065223444609487328, 5.59768319690431338580334897316, 6.97092242988088816256993614523, 7.86063465600745783264813086438, 9.035718570315973199417007272758, 9.368368686525530344826145041238, 10.50795284610599784451485811020

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