Properties

Label 2-289800-1.1-c1-0-98
Degree $2$
Conductor $289800$
Sign $-1$
Analytic cond. $2314.06$
Root an. cond. $48.1047$
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  + 7-s + 3·11-s − 3·13-s + 3·17-s − 2·19-s + 23-s − 3·29-s − 4·31-s − 8·37-s + 10·41-s − 4·43-s + 13·47-s + 49-s + 8·59-s − 10·61-s + 6·67-s − 6·73-s + 3·77-s + 7·79-s + 10·89-s − 3·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.377·7-s + 0.904·11-s − 0.832·13-s + 0.727·17-s − 0.458·19-s + 0.208·23-s − 0.557·29-s − 0.718·31-s − 1.31·37-s + 1.56·41-s − 0.609·43-s + 1.89·47-s + 1/7·49-s + 1.04·59-s − 1.28·61-s + 0.733·67-s − 0.702·73-s + 0.341·77-s + 0.787·79-s + 1.05·89-s − 0.314·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(2314.06\)
Root analytic conductor: \(48.1047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{289800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 289800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
23 \( 1 - T \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−12.80946747776016, −12.36572252418641, −12.15555141348693, −11.58942913364136, −11.15389694053499, −10.61715678458689, −10.28273384729405, −9.676490726616234, −9.180446762459438, −8.923813290120543, −8.359304437312407, −7.678339246528228, −7.412912354870260, −6.956240184957077, −6.348498622220847, −5.846705353714496, −5.317455790097322, −4.918065084919599, −4.228610756235180, −3.832560123780629, −3.326660010651012, −2.544040996023415, −2.097968469867870, −1.436071652960787, −0.8371127942833319, 0, 0.8371127942833319, 1.436071652960787, 2.097968469867870, 2.544040996023415, 3.326660010651012, 3.832560123780629, 4.228610756235180, 4.918065084919599, 5.317455790097322, 5.846705353714496, 6.348498622220847, 6.956240184957077, 7.412912354870260, 7.678339246528228, 8.359304437312407, 8.923813290120543, 9.180446762459438, 9.676490726616234, 10.28273384729405, 10.61715678458689, 11.15389694053499, 11.58942913364136, 12.15555141348693, 12.36572252418641, 12.80946747776016

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