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Properties

Label 2-323400-1.1-c1-0-130
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 − 2·13-s + 2·17-s + 4·19-s + 27-s − 2·29-s + 8·31-s + 33-s + 2·37-s − 2·39-s + 6·41-s − 4·43-s + 8·47-s + 2·51-s + 10·53-s + 4·57-s + 4·59-s + 2·61-s + 12·67-s + 8·71-s − 14·73-s + 8·79-s + 81-s + 12·83-s − 2·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 1.16·47-s + 0.280·51-s + 1.37·53-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 1.46·67-s + 0.949·71-s − 1.63·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.214·87-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

\(L(1)\) \(\approx\) \(4.868965356\)
\(L(\frac12)\) \(\approx\) \(4.868965356\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.51350816683457, −12.22181255339175, −11.74596510222009, −11.36491340459664, −10.72616265257515, −10.24455377164922, −9.789209558568743, −9.489205069057154, −8.995144233633978, −8.442566865408438, −8.009910870454392, −7.580770832586466, −7.086074452134012, −6.699286497480010, −6.036997538912086, −5.522979101674153, −5.073476147766062, −4.461199305055897, −3.959826032872171, −3.481679301108089, −2.847921382930798, −2.434326493419410, −1.843632841882683, −0.9875339306455484, −0.6626145584888935, 0.6626145584888935, 0.9875339306455484, 1.843632841882683, 2.434326493419410, 2.847921382930798, 3.481679301108089, 3.959826032872171, 4.461199305055897, 5.073476147766062, 5.522979101674153, 6.036997538912086, 6.699286497480010, 7.086074452134012, 7.580770832586466, 8.009910870454392, 8.442566865408438, 8.995144233633978, 9.489205069057154, 9.789209558568743, 10.24455377164922, 10.72616265257515, 11.36491340459664, 11.74596510222009, 12.22181255339175, 12.51350816683457

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