Properties

Label 2-123840-1.1-c1-0-120
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 + 4·7-s + 2·13-s − 2·17-s − 4·19-s + 25-s + 2·29-s − 4·35-s − 10·37-s + 6·41-s + 43-s + 8·47-s + 9·49-s + 2·53-s − 2·61-s − 2·65-s + 4·67-s − 8·71-s + 10·73-s − 8·79-s + 16·83-s + 2·85-s − 6·89-s + 8·91-s + 4·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s − 0.676·35-s − 1.64·37-s + 0.937·41-s + 0.152·43-s + 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.256·61-s − 0.248·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.900·79-s + 1.75·83-s + 0.216·85-s − 0.635·89-s + 0.838·91-s + 0.410·95-s + 0.203·97-s + 0.0995·101-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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
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 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.89767898956441, −13.38173244472517, −12.71363907390979, −12.27598339487558, −11.82959454044330, −11.31278591438995, −10.83490469598836, −10.64621731061887, −10.03440528087575, −9.148841104930085, −8.779047802901478, −8.452305802613782, −7.823820004581707, −7.539296076114293, −6.795982444263490, −6.388912706235693, −5.635716891787019, −5.151354765872050, −4.634485108100675, −4.053944655665463, −3.743385300482935, −2.758108784022960, −2.215529886931365, −1.563639455226136, −0.9543272715444760, 0, 0.9543272715444760, 1.563639455226136, 2.215529886931365, 2.758108784022960, 3.743385300482935, 4.053944655665463, 4.634485108100675, 5.151354765872050, 5.635716891787019, 6.388912706235693, 6.795982444263490, 7.539296076114293, 7.823820004581707, 8.452305802613782, 8.779047802901478, 9.148841104930085, 10.03440528087575, 10.64621731061887, 10.83490469598836, 11.31278591438995, 11.82959454044330, 12.27598339487558, 12.71363907390979, 13.38173244472517, 13.89767898956441

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