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Properties

Label 2-462e2-1.1-c1-0-32
Degree $2$
Conductor $213444$
Sign $1$
Analytic cond. $1704.35$
Root an. cond. $41.2838$
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  − 3·5-s + 5·13-s + 6·17-s − 19-s + 4·25-s + 9·29-s − 8·31-s − 7·37-s + 6·41-s + 10·43-s − 3·47-s + 6·53-s + 9·59-s + 2·61-s − 15·65-s − 13·67-s + 6·71-s + 11·73-s + 10·79-s − 6·83-s − 18·85-s − 6·89-s + 3·95-s + 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.38·13-s + 1.45·17-s − 0.229·19-s + 4/5·25-s + 1.67·29-s − 1.43·31-s − 1.15·37-s + 0.937·41-s + 1.52·43-s − 0.437·47-s + 0.824·53-s + 1.17·59-s + 0.256·61-s − 1.86·65-s − 1.58·67-s + 0.712·71-s + 1.28·73-s + 1.12·79-s − 0.658·83-s − 1.95·85-s − 0.635·89-s + 0.307·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1704.35\)
Root analytic conductor: \(41.2838\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 213444,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.608887362\)
\(L(\frac12)\) \(\approx\) \(2.608887362\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.83748324177356, −12.48312722516758, −12.02798610833276, −11.71270011668639, −11.02261151985175, −10.84863430670545, −10.30725836696467, −9.769878735688112, −9.052445961500937, −8.726511460208751, −8.145941634182235, −7.917822943705560, −7.287136457007808, −6.928748755162154, −6.257155771174567, −5.705454270777504, −5.319214382551094, −4.518820105763047, −4.110401130193605, −3.471593030560561, −3.377324505324293, −2.528694920536466, −1.724939132892777, −0.9362234275867142, −0.5806280233031980, 0.5806280233031980, 0.9362234275867142, 1.724939132892777, 2.528694920536466, 3.377324505324293, 3.471593030560561, 4.110401130193605, 4.518820105763047, 5.319214382551094, 5.705454270777504, 6.257155771174567, 6.928748755162154, 7.287136457007808, 7.917822943705560, 8.145941634182235, 8.726511460208751, 9.052445961500937, 9.769878735688112, 10.30725836696467, 10.84863430670545, 11.02261151985175, 11.71270011668639, 12.02798610833276, 12.48312722516758, 12.83748324177356

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