Properties

Degree $2$
Conductor $26010$
Sign $-1$
Motivic weight $1$
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 + 4·7-s + 8-s − 10-s + 2·13-s + 4·14-s + 16-s − 4·19-s − 20-s + 25-s + 2·26-s + 4·28-s − 6·29-s − 8·31-s + 32-s − 4·35-s − 2·37-s − 4·38-s − 40-s − 6·41-s − 4·43-s + 9·49-s + 50-s + 2·52-s + 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.755·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.676·35-s − 0.328·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 9/7·49-s + 0.141·50-s + 0.277·52-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26010\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{26010} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 26010,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
17 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.33710350484107, −14.93084989818551, −14.63504078049401, −14.07423139950132, −13.42145125099919, −12.97939894737233, −12.37691764853676, −11.72978122313588, −11.34093469713616, −10.90359102773012, −10.45625279354257, −9.610618551250649, −8.743505261608749, −8.389759028109602, −7.864092183739410, −7.106591554319463, −6.790733154333101, −5.644071550836131, −5.496073758243726, −4.682098758245274, −4.077927821699721, −3.673646414171904, −2.702348932863948, −1.839955897595660, −1.398712384968503, 0, 1.398712384968503, 1.839955897595660, 2.702348932863948, 3.673646414171904, 4.077927821699721, 4.682098758245274, 5.496073758243726, 5.644071550836131, 6.790733154333101, 7.106591554319463, 7.864092183739410, 8.389759028109602, 8.743505261608749, 9.610618551250649, 10.45625279354257, 10.90359102773012, 11.34093469713616, 11.72978122313588, 12.37691764853676, 12.97939894737233, 13.42145125099919, 14.07423139950132, 14.63504078049401, 14.93084989818551, 15.33710350484107

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