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Properties

Label 2-323400-1.1-c1-0-101
Degree $2$
Conductor $323400$
Sign $1$
Analytic cond. $2582.36$
Root an. cond. $50.8169$
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-s + 9-s − 11-s − 5·13-s − 17-s − 19-s + 8·23-s + 27-s + 6·31-s − 33-s + 7·37-s − 5·39-s + 41-s − 6·43-s − 2·47-s − 51-s + 9·53-s − 57-s + 14·59-s + 3·61-s + 5·67-s + 8·69-s + 71-s − 7·73-s + 8·79-s + 81-s + 8·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.242·17-s − 0.229·19-s + 1.66·23-s + 0.192·27-s + 1.07·31-s − 0.174·33-s + 1.15·37-s − 0.800·39-s + 0.156·41-s − 0.914·43-s − 0.291·47-s − 0.140·51-s + 1.23·53-s − 0.132·57-s + 1.82·59-s + 0.384·61-s + 0.610·67-s + 0.963·69-s + 0.118·71-s − 0.819·73-s + 0.900·79-s + 1/9·81-s + 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(323400\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(2582.36\)
Root analytic conductor: \(50.8169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 323400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.837135716\)
\(L(\frac12)\) \(\approx\) \(3.837135716\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−12.87876070948082, −12.14065763118774, −11.70505255935704, −11.36744403881799, −10.68712942809944, −10.28016639465296, −9.834307215845592, −9.494630659879183, −8.839521805133429, −8.608679555138977, −7.957404422280603, −7.557731359580173, −7.106706913856277, −6.650219190702725, −6.196927987402063, −5.344609177245850, −5.043203512418326, −4.571523928947801, −4.057528008876085, −3.305427789866913, −2.932782690083912, −2.257152750693288, −2.071621925747630, −0.9470681533251791, −0.5851230881873696, 0.5851230881873696, 0.9470681533251791, 2.071621925747630, 2.257152750693288, 2.932782690083912, 3.305427789866913, 4.057528008876085, 4.571523928947801, 5.043203512418326, 5.344609177245850, 6.196927987402063, 6.650219190702725, 7.106706913856277, 7.557731359580173, 7.957404422280603, 8.608679555138977, 8.839521805133429, 9.494630659879183, 9.834307215845592, 10.28016639465296, 10.68712942809944, 11.36744403881799, 11.70505255935704, 12.14065763118774, 12.87876070948082

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