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Properties

Label 2-330e2-1.1-c1-0-7
Degree $2$
Conductor $108900$
Sign $1$
Analytic cond. $869.570$
Root an. cond. $29.4884$
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  − 3·7-s − 5·13-s − 2·17-s − 4·19-s + 4·23-s + 4·29-s + 8·31-s − 7·37-s − 12·41-s − 5·43-s + 8·47-s + 2·49-s + 14·53-s − 4·59-s − 6·61-s + 12·67-s + 6·71-s − 5·73-s + 3·79-s + 10·83-s − 12·89-s + 15·91-s − 97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.13·7-s − 1.38·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 0.742·29-s + 1.43·31-s − 1.15·37-s − 1.87·41-s − 0.762·43-s + 1.16·47-s + 2/7·49-s + 1.92·53-s − 0.520·59-s − 0.768·61-s + 1.46·67-s + 0.712·71-s − 0.585·73-s + 0.337·79-s + 1.09·83-s − 1.27·89-s + 1.57·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(869.570\)
Root analytic conductor: \(29.4884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 108900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6183794499\)
\(L(\frac12)\) \(\approx\) \(0.6183794499\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.64190813629474, −13.22476234871843, −12.64291260647781, −12.09275444615092, −12.05013821372717, −11.23620880238203, −10.53508167423247, −10.21876059308017, −9.859381814033791, −9.244450891365868, −8.727163382404763, −8.341800016864723, −7.630236985185642, −6.947802850518115, −6.660913898264414, −6.361228778871816, −5.369933288057175, −5.075579558589507, −4.420379963714600, −3.822939607689189, −3.144917361293646, −2.609849966075429, −2.154189127974579, −1.169265596747302, −0.2529281990164607, 0.2529281990164607, 1.169265596747302, 2.154189127974579, 2.609849966075429, 3.144917361293646, 3.822939607689189, 4.420379963714600, 5.075579558589507, 5.369933288057175, 6.361228778871816, 6.660913898264414, 6.947802850518115, 7.630236985185642, 8.341800016864723, 8.727163382404763, 9.244450891365868, 9.859381814033791, 10.21876059308017, 10.53508167423247, 11.23620880238203, 12.05013821372717, 12.09275444615092, 12.64291260647781, 13.22476234871843, 13.64190813629474

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