Properties

Label 2-123840-1.1-c1-0-82
Degree $2$
Conductor $123840$
Sign $-1$
Analytic cond. $988.867$
Root an. cond. $31.4462$
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  − 5-s + 7-s − 7·13-s + 4·17-s − 19-s + 25-s − 29-s + 3·31-s − 35-s + 8·37-s − 9·41-s + 43-s − 10·47-s − 6·49-s + 14·53-s − 5·61-s + 7·65-s − 11·67-s + 10·71-s + 7·73-s + 13·79-s − 8·83-s − 4·85-s + 12·89-s − 7·91-s + 95-s + 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 1.94·13-s + 0.970·17-s − 0.229·19-s + 1/5·25-s − 0.185·29-s + 0.538·31-s − 0.169·35-s + 1.31·37-s − 1.40·41-s + 0.152·43-s − 1.45·47-s − 6/7·49-s + 1.92·53-s − 0.640·61-s + 0.868·65-s − 1.34·67-s + 1.18·71-s + 0.819·73-s + 1.46·79-s − 0.878·83-s − 0.433·85-s + 1.27·89-s − 0.733·91-s + 0.102·95-s + 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(123840\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $-1$
Analytic conductor: \(988.867\)
Root analytic conductor: \(31.4462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{123840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 123840,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
43 \( 1 - T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.79525016341386, −13.25934188735257, −12.74685774267755, −12.19091678860267, −11.91833428769566, −11.51224114768525, −10.91165971383142, −10.18190285388446, −10.05045597464141, −9.411335979865307, −8.966375353481826, −8.111076038186750, −7.908783674836635, −7.505866642042541, −6.787032225300702, −6.476454171605979, −5.578900933082432, −5.130592999204526, −4.723789930778515, −4.155026193897734, −3.454709543703420, −2.866022710624240, −2.304862512477215, −1.616347695203834, −0.7677882734084680, 0, 0.7677882734084680, 1.616347695203834, 2.304862512477215, 2.866022710624240, 3.454709543703420, 4.155026193897734, 4.723789930778515, 5.130592999204526, 5.578900933082432, 6.476454171605979, 6.787032225300702, 7.505866642042541, 7.908783674836635, 8.111076038186750, 8.966375353481826, 9.411335979865307, 10.05045597464141, 10.18190285388446, 10.91165971383142, 11.51224114768525, 11.91833428769566, 12.19091678860267, 12.74685774267755, 13.25934188735257, 13.79525016341386

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