Properties

Label 2-323400-1.1-c1-0-10
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 − 4·13-s − 6·23-s + 27-s − 2·29-s − 33-s − 4·37-s − 4·39-s + 6·41-s + 4·43-s − 4·47-s − 6·53-s − 10·61-s + 10·67-s − 6·69-s − 8·71-s + 14·73-s + 8·79-s + 81-s − 2·87-s − 8·89-s − 10·97-s − 99-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 1.25·23-s + 0.192·27-s − 0.371·29-s − 0.174·33-s − 0.657·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.583·47-s − 0.824·53-s − 1.28·61-s + 1.22·67-s − 0.722·69-s − 0.949·71-s + 1.63·73-s + 0.900·79-s + 1/9·81-s − 0.214·87-s − 0.847·89-s − 1.01·97-s − 0.100·99-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(1.059546494\)
\(L(\frac12)\) \(\approx\) \(1.059546494\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−12.70359678463234, −12.16332923384968, −11.88285216532388, −11.20717683323254, −10.69108451977687, −10.39089261750529, −9.696813912816659, −9.457583501802168, −9.137267592739715, −8.255467956484979, −8.064074239417755, −7.661657998274213, −7.076182490764417, −6.668530864907088, −6.070364886161489, −5.487327238471838, −5.075873765640068, −4.441763149051091, −4.030492518532363, −3.465485486038949, −2.808145616338578, −2.401350216755055, −1.862413349233005, −1.222701594932522, −0.2524719241516798, 0.2524719241516798, 1.222701594932522, 1.862413349233005, 2.401350216755055, 2.808145616338578, 3.465485486038949, 4.030492518532363, 4.441763149051091, 5.075873765640068, 5.487327238471838, 6.070364886161489, 6.668530864907088, 7.076182490764417, 7.661657998274213, 8.064074239417755, 8.255467956484979, 9.137267592739715, 9.457583501802168, 9.696813912816659, 10.39089261750529, 10.69108451977687, 11.20717683323254, 11.88285216532388, 12.16332923384968, 12.70359678463234

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