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Properties

Label 2-510e2-1.1-c1-0-18
Degree $2$
Conductor $260100$
Sign $1$
Analytic cond. $2076.90$
Root an. cond. $45.5731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 3·11-s + 4·13-s + 19-s + 2·23-s − 9·29-s + 4·31-s − 11·37-s + 7·41-s + 6·43-s − 13·47-s − 6·49-s − 11·53-s + 14·59-s + 14·61-s − 4·67-s + 8·71-s − 73-s + 3·77-s + 14·79-s + 12·83-s − 10·89-s − 4·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.904·11-s + 1.10·13-s + 0.229·19-s + 0.417·23-s − 1.67·29-s + 0.718·31-s − 1.80·37-s + 1.09·41-s + 0.914·43-s − 1.89·47-s − 6/7·49-s − 1.51·53-s + 1.82·59-s + 1.79·61-s − 0.488·67-s + 0.949·71-s − 0.117·73-s + 0.341·77-s + 1.57·79-s + 1.31·83-s − 1.05·89-s − 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2076.90\)
Root analytic conductor: \(45.5731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 260100,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.532687238\)
\(L(\frac12)\) \(\approx\) \(1.532687238\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−12.88465582403183, −12.51907912458264, −11.86542058862568, −11.25773701032260, −11.05295827687110, −10.60575824719730, −9.995533555640344, −9.574864639982220, −9.180511958522439, −8.567265548888458, −8.081659926897078, −7.822474251247929, −7.067813283337780, −6.694872488190739, −6.199283575194582, −5.584056315652016, −5.217098409768874, −4.744260995742900, −3.844580794767288, −3.631837508437754, −3.034686702070177, −2.399755047836909, −1.794775127927744, −1.125705900579431, −0.3540894466954424, 0.3540894466954424, 1.125705900579431, 1.794775127927744, 2.399755047836909, 3.034686702070177, 3.631837508437754, 3.844580794767288, 4.744260995742900, 5.217098409768874, 5.584056315652016, 6.199283575194582, 6.694872488190739, 7.067813283337780, 7.822474251247929, 8.081659926897078, 8.567265548888458, 9.180511958522439, 9.574864639982220, 9.995533555640344, 10.60575824719730, 11.05295827687110, 11.25773701032260, 11.86542058862568, 12.51907912458264, 12.88465582403183

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