Properties

Label 2-5290-1.1-c1-0-3
Degree $2$
Conductor $5290$
Sign $1$
Analytic cond. $42.2408$
Root an. cond. $6.49929$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.11·3-s + 4-s + 5-s − 3.11·6-s − 4.50·7-s + 8-s + 6.72·9-s + 10-s − 4.33·11-s − 3.11·12-s − 3.72·13-s − 4.50·14-s − 3.11·15-s + 16-s − 1.11·17-s + 6.72·18-s − 4.50·19-s + 20-s + 14.0·21-s − 4.33·22-s − 3.11·24-s + 25-s − 3.72·26-s − 11.6·27-s − 4.50·28-s − 8.23·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.447·5-s − 1.27·6-s − 1.70·7-s + 0.353·8-s + 2.24·9-s + 0.316·10-s − 1.30·11-s − 0.900·12-s − 1.03·13-s − 1.20·14-s − 0.805·15-s + 0.250·16-s − 0.271·17-s + 1.58·18-s − 1.03·19-s + 0.223·20-s + 3.06·21-s − 0.924·22-s − 0.636·24-s + 0.200·25-s − 0.731·26-s − 2.23·27-s − 0.852·28-s − 1.52·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5290\)    =    \(2 \cdot 5 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(42.2408\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5290,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3758378563\)
\(L(\frac12)\) \(\approx\) \(0.3758378563\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + 3.11T + 3T^{2} \)
7 \( 1 + 4.50T + 7T^{2} \)
11 \( 1 + 4.33T + 11T^{2} \)
13 \( 1 + 3.72T + 13T^{2} \)
17 \( 1 + 1.11T + 17T^{2} \)
19 \( 1 + 4.50T + 19T^{2} \)
29 \( 1 + 8.23T + 29T^{2} \)
31 \( 1 - 1.72T + 31T^{2} \)
37 \( 1 - 0.781T + 37T^{2} \)
41 \( 1 - 3.90T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 2.23T + 59T^{2} \)
61 \( 1 + 3.55T + 61T^{2} \)
67 \( 1 + 2.43T + 67T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + 9.45T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 2.78T + 83T^{2} \)
89 \( 1 - 7.69T + 89T^{2} \)
97 \( 1 - 0.642T + 97T^{2} \)
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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

−7.80291975384119026541036630692, −7.05800244529518647377369516779, −6.48352626563470051816625721910, −6.03222030876671724833400775412, −5.31998746870685384338984166895, −4.83721550610330969470009775061, −3.91010174237304585126725868440, −2.88157900574706641779196063538, −1.97621079577623223500824879106, −0.30239563090186820739142845946, 0.30239563090186820739142845946, 1.97621079577623223500824879106, 2.88157900574706641779196063538, 3.91010174237304585126725868440, 4.83721550610330969470009775061, 5.31998746870685384338984166895, 6.03222030876671724833400775412, 6.48352626563470051816625721910, 7.05800244529518647377369516779, 7.80291975384119026541036630692

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