Properties

Label 2-323400-1.1-c1-0-129
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 11-s − 3·13-s + 4·17-s − 19-s − 2·23-s − 27-s − 29-s − 4·31-s + 33-s + 3·37-s + 3·39-s − 4·41-s − 2·43-s + 5·47-s − 4·51-s − 2·53-s + 57-s − 9·59-s + 2·61-s + 3·67-s + 2·69-s + 12·71-s + 7·73-s − 8·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.970·17-s − 0.229·19-s − 0.417·23-s − 0.192·27-s − 0.185·29-s − 0.718·31-s + 0.174·33-s + 0.493·37-s + 0.480·39-s − 0.624·41-s − 0.304·43-s + 0.729·47-s − 0.560·51-s − 0.274·53-s + 0.132·57-s − 1.17·59-s + 0.256·61-s + 0.366·67-s + 0.240·69-s + 1.42·71-s + 0.819·73-s − 0.900·79-s + 1/9·81-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: \(1\)
Selberg data: \((2,\ 323400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.90757332191871, −12.27623955902151, −12.01616731062932, −11.53260459120277, −11.03417319139294, −10.47247362340614, −10.28561149462192, −9.562777729781201, −9.430459956962298, −8.733425311095331, −8.108289805703019, −7.679313852737722, −7.413977950973864, −6.687927375722754, −6.357809613740229, −5.754009044959167, −5.195065485436791, −5.042416643833795, −4.289618833569675, −3.784470115687064, −3.253290473424743, −2.566825408769767, −2.041347515268345, −1.398186702256106, −0.6591492494238998, 0, 0.6591492494238998, 1.398186702256106, 2.041347515268345, 2.566825408769767, 3.253290473424743, 3.784470115687064, 4.289618833569675, 5.042416643833795, 5.195065485436791, 5.754009044959167, 6.357809613740229, 6.687927375722754, 7.413977950973864, 7.679313852737722, 8.108289805703019, 8.733425311095331, 9.430459956962298, 9.562777729781201, 10.28561149462192, 10.47247362340614, 11.03417319139294, 11.53260459120277, 12.01616731062932, 12.27623955902151, 12.90757332191871

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