Properties

Label 2-507-1.1-c3-0-39
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.12·2-s − 3·3-s + 9·4-s + 13.4·5-s + 12.3·6-s − 31.4·7-s − 4.12·8-s + 9·9-s − 55.4·10-s + 40.4·11-s − 27·12-s + 129.·14-s − 40.3·15-s − 55·16-s − 43.1·17-s − 37.1·18-s − 26.9·19-s + 120.·20-s + 94.2·21-s − 166.·22-s − 19.0·23-s + 12.3·24-s + 55.6·25-s − 27·27-s − 282.·28-s − 154.·29-s + 166.·30-s + ⋯
L(s)  = 1  − 1.45·2-s − 0.577·3-s + 1.12·4-s + 1.20·5-s + 0.841·6-s − 1.69·7-s − 0.182·8-s + 0.333·9-s − 1.75·10-s + 1.11·11-s − 0.649·12-s + 2.47·14-s − 0.694·15-s − 0.859·16-s − 0.615·17-s − 0.485·18-s − 0.325·19-s + 1.35·20-s + 0.979·21-s − 1.61·22-s − 0.172·23-s + 0.105·24-s + 0.445·25-s − 0.192·27-s − 1.90·28-s − 0.986·29-s + 1.01·30-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: \(1\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.12T + 8T^{2} \)
5 \( 1 - 13.4T + 125T^{2} \)
7 \( 1 + 31.4T + 343T^{2} \)
11 \( 1 - 40.4T + 1.33e3T^{2} \)
17 \( 1 + 43.1T + 4.91e3T^{2} \)
19 \( 1 + 26.9T + 6.85e3T^{2} \)
23 \( 1 + 19.0T + 1.21e4T^{2} \)
29 \( 1 + 154.T + 2.43e4T^{2} \)
31 \( 1 - 308.T + 2.97e4T^{2} \)
37 \( 1 - 43.5T + 5.06e4T^{2} \)
41 \( 1 - 47.8T + 6.89e4T^{2} \)
43 \( 1 - 342.T + 7.95e4T^{2} \)
47 \( 1 - 133.T + 1.03e5T^{2} \)
53 \( 1 + 438.T + 1.48e5T^{2} \)
59 \( 1 - 590.T + 2.05e5T^{2} \)
61 \( 1 + 541.T + 2.26e5T^{2} \)
67 \( 1 + 230.T + 3.00e5T^{2} \)
71 \( 1 - 449.T + 3.57e5T^{2} \)
73 \( 1 + 389.T + 3.89e5T^{2} \)
79 \( 1 + 897.T + 4.93e5T^{2} \)
83 \( 1 + 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + 925.T + 7.04e5T^{2} \)
97 \( 1 + 1.56e3T + 9.12e5T^{2} \)
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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

−9.755624259469251111108886627608, −9.480196045044593693961056231165, −8.655896913034123005434833329213, −7.18865853427112873591183271993, −6.44052118598353183017102799416, −5.93187246775876670481689550050, −4.20841270190841640600250909533, −2.55472301057076868574984331212, −1.27269060929327938250794938671, 0, 1.27269060929327938250794938671, 2.55472301057076868574984331212, 4.20841270190841640600250909533, 5.93187246775876670481689550050, 6.44052118598353183017102799416, 7.18865853427112873591183271993, 8.655896913034123005434833329213, 9.480196045044593693961056231165, 9.755624259469251111108886627608

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