Properties

Label 2-5265-1.1-c1-0-100
Degree $2$
Conductor $5265$
Sign $-1$
Analytic cond. $42.0412$
Root an. cond. $6.48392$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 5-s − 7-s + 3·8-s + 10-s + 2·11-s − 13-s + 14-s − 16-s − 4·17-s + 20-s − 2·22-s − 3·23-s + 25-s + 26-s + 28-s − 29-s + 8·31-s − 5·32-s + 4·34-s + 35-s + 4·37-s − 3·40-s + 9·41-s − 8·43-s − 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s + 1.06·8-s + 0.316·10-s + 0.603·11-s − 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 0.185·29-s + 1.43·31-s − 0.883·32-s + 0.685·34-s + 0.169·35-s + 0.657·37-s − 0.474·40-s + 1.40·41-s − 1.21·43-s − 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5265\)    =    \(3^{4} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(42.0412\)
Root analytic conductor: \(6.48392\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5265,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.967833033256388860983506040528, −7.29400997155652741184575997307, −6.53974007232656671800856146389, −5.78723750908105204111581388099, −4.57533895512732778965938982892, −4.32980124235823285316453228548, −3.32464586839880143522492712406, −2.23113386598186817711149388649, −1.05860211671937643906938193219, 0, 1.05860211671937643906938193219, 2.23113386598186817711149388649, 3.32464586839880143522492712406, 4.32980124235823285316453228548, 4.57533895512732778965938982892, 5.78723750908105204111581388099, 6.53974007232656671800856146389, 7.29400997155652741184575997307, 7.967833033256388860983506040528

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