Properties

Label 2-507-1.1-c3-0-50
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  + 4.03·2-s + 3·3-s + 8.24·4-s + 8.08·5-s + 12.0·6-s + 5.95·7-s + 0.978·8-s + 9·9-s + 32.5·10-s + 17.2·11-s + 24.7·12-s + 23.9·14-s + 24.2·15-s − 61.9·16-s + 92.9·17-s + 36.2·18-s + 13.3·19-s + 66.6·20-s + 17.8·21-s + 69.4·22-s + 219.·23-s + 2.93·24-s − 59.5·25-s + 27·27-s + 49.0·28-s − 199.·29-s + 97.7·30-s + ⋯
L(s)  = 1  + 1.42·2-s + 0.577·3-s + 1.03·4-s + 0.723·5-s + 0.822·6-s + 0.321·7-s + 0.0432·8-s + 0.333·9-s + 1.03·10-s + 0.472·11-s + 0.594·12-s + 0.457·14-s + 0.417·15-s − 0.968·16-s + 1.32·17-s + 0.474·18-s + 0.161·19-s + 0.745·20-s + 0.185·21-s + 0.673·22-s + 1.99·23-s + 0.0249·24-s − 0.476·25-s + 0.192·27-s + 0.331·28-s − 1.27·29-s + 0.595·30-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

\(L(2)\) \(\approx\) \(6.208951717\)
\(L(\frac12)\) \(\approx\) \(6.208951717\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.03T + 8T^{2} \)
5 \( 1 - 8.08T + 125T^{2} \)
7 \( 1 - 5.95T + 343T^{2} \)
11 \( 1 - 17.2T + 1.33e3T^{2} \)
17 \( 1 - 92.9T + 4.91e3T^{2} \)
19 \( 1 - 13.3T + 6.85e3T^{2} \)
23 \( 1 - 219.T + 1.21e4T^{2} \)
29 \( 1 + 199.T + 2.43e4T^{2} \)
31 \( 1 - 307.T + 2.97e4T^{2} \)
37 \( 1 + 333.T + 5.06e4T^{2} \)
41 \( 1 + 200.T + 6.89e4T^{2} \)
43 \( 1 - 116.T + 7.95e4T^{2} \)
47 \( 1 - 338.T + 1.03e5T^{2} \)
53 \( 1 + 26.6T + 1.48e5T^{2} \)
59 \( 1 + 280.T + 2.05e5T^{2} \)
61 \( 1 + 207.T + 2.26e5T^{2} \)
67 \( 1 + 285.T + 3.00e5T^{2} \)
71 \( 1 + 317.T + 3.57e5T^{2} \)
73 \( 1 + 63.0T + 3.89e5T^{2} \)
79 \( 1 + 623.T + 4.93e5T^{2} \)
83 \( 1 - 659.T + 5.71e5T^{2} \)
89 \( 1 - 1.27e3T + 7.04e5T^{2} \)
97 \( 1 - 603.T + 9.12e5T^{2} \)
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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

−10.61565739024240660949416625759, −9.565916868868031322316607131741, −8.825835484782599119631179628435, −7.56709047841572591635482857706, −6.57438940273698134261969615436, −5.58420071729359679427479207154, −4.83662252958104961951091298664, −3.64924417508969783512614572676, −2.80596400132224719434126670634, −1.46911157111870841040347983419, 1.46911157111870841040347983419, 2.80596400132224719434126670634, 3.64924417508969783512614572676, 4.83662252958104961951091298664, 5.58420071729359679427479207154, 6.57438940273698134261969615436, 7.56709047841572591635482857706, 8.825835484782599119631179628435, 9.565916868868031322316607131741, 10.61565739024240660949416625759

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