Properties

Label 2-323400-1.1-c1-0-124
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 − 4·13-s − 7·17-s − 4·23-s + 27-s − 29-s − 2·31-s − 33-s + 3·37-s − 4·39-s + 7·41-s − 3·43-s − 7·47-s − 7·51-s + 2·53-s − 12·59-s + 4·61-s − 2·67-s − 4·69-s + 15·71-s + 2·73-s + 11·79-s + 81-s + 2·83-s − 87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 1.69·17-s − 0.834·23-s + 0.192·27-s − 0.185·29-s − 0.359·31-s − 0.174·33-s + 0.493·37-s − 0.640·39-s + 1.09·41-s − 0.457·43-s − 1.02·47-s − 0.980·51-s + 0.274·53-s − 1.56·59-s + 0.512·61-s − 0.244·67-s − 0.481·69-s + 1.78·71-s + 0.234·73-s + 1.23·79-s + 1/9·81-s + 0.219·83-s − 0.107·87-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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−12.94733756285854, −12.42079878466633, −12.02752744804970, −11.46927348740167, −10.86604615716249, −10.74701481558862, −9.949423436973404, −9.581863829996633, −9.297785997860968, −8.710360008985074, −8.220578063017592, −7.773684119506654, −7.419473933538952, −6.740971360241203, −6.478561897791245, −5.854026253307253, −5.127970160827581, −4.829550668848664, −4.174300044055459, −3.880711557238057, −3.053856960678090, −2.600998892321192, −2.085644956815603, −1.700990142489901, −0.6468697496856385, 0, 0.6468697496856385, 1.700990142489901, 2.085644956815603, 2.600998892321192, 3.053856960678090, 3.880711557238057, 4.174300044055459, 4.829550668848664, 5.127970160827581, 5.854026253307253, 6.478561897791245, 6.740971360241203, 7.419473933538952, 7.773684119506654, 8.220578063017592, 8.710360008985074, 9.297785997860968, 9.581863829996633, 9.949423436973404, 10.74701481558862, 10.86604615716249, 11.46927348740167, 12.02752744804970, 12.42079878466633, 12.94733756285854

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