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Properties

Label 2-82810-1.1-c1-0-65
Degree $2$
Conductor $82810$
Sign $-1$
Analytic cond. $661.241$
Root an. cond. $25.7146$
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 + 2·3-s + 4-s + 5-s − 2·6-s − 8-s + 9-s − 10-s + 3·11-s + 2·12-s + 2·15-s + 16-s + 6·17-s − 18-s − 5·19-s + 20-s − 3·22-s − 2·24-s + 25-s − 4·27-s − 2·30-s + 4·31-s − 32-s + 6·33-s − 6·34-s + 36-s + 11·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.577·12-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.639·22-s − 0.408·24-s + 1/5·25-s − 0.769·27-s − 0.365·30-s + 0.718·31-s − 0.176·32-s + 1.04·33-s − 1.02·34-s + 1/6·36-s + 1.80·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82810\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(661.241\)
Root analytic conductor: \(25.7146\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 82810,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.46485967059545, −13.73273785381669, −13.35258943123835, −12.76046267224398, −12.14873831406919, −11.78383194712775, −11.11417182641173, −10.59075168380827, −9.911410990667284, −9.725361286395692, −9.091743906809239, −8.727372265330651, −8.251061529100341, −7.696641918449462, −7.344123076546927, −6.510140426717330, −6.093494969254101, −5.640626964389328, −4.639720698791703, −4.141413488776931, −3.385954326679340, −2.896206990844316, −2.381706886868168, −1.552619737288039, −1.186172077487330, 0, 1.186172077487330, 1.552619737288039, 2.381706886868168, 2.896206990844316, 3.385954326679340, 4.141413488776931, 4.639720698791703, 5.640626964389328, 6.093494969254101, 6.510140426717330, 7.344123076546927, 7.696641918449462, 8.251061529100341, 8.727372265330651, 9.091743906809239, 9.725361286395692, 9.911410990667284, 10.59075168380827, 11.11417182641173, 11.78383194712775, 12.14873831406919, 12.76046267224398, 13.35258943123835, 13.73273785381669, 14.46485967059545

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