Properties

Degree $2$
Conductor $116380$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s + 4·7-s + 9-s + 11-s − 4·13-s + 2·15-s + 4·19-s − 8·21-s + 25-s + 4·27-s − 6·29-s + 8·31-s − 2·33-s − 4·35-s − 2·37-s + 8·39-s + 6·41-s − 8·43-s − 45-s + 6·47-s + 9·49-s + 6·53-s − 55-s − 8·57-s − 12·59-s − 2·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.516·15-s + 0.917·19-s − 1.74·21-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 1.43·31-s − 0.348·33-s − 0.676·35-s − 0.328·37-s + 1.28·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s − 1.05·57-s − 1.56·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116380\)    =    \(2^{2} \cdot 5 \cdot 11 \cdot 23^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{116380} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 116380,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.152587269\)
\(L(\frac12)\) \(\approx\) \(1.152587269\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−13.72582329503388, −12.99627605409502, −12.32089961728670, −12.00228016720157, −11.66169848830913, −11.32403705532243, −10.83984637092712, −10.32361441493818, −9.836634857209514, −9.163377049416901, −8.640400140832530, −8.015139192768306, −7.655106326071043, −7.094075861410001, −6.693602839251429, −5.788059683546901, −5.512175547564607, −5.029127772418323, −4.355649316657311, −4.269355287127342, −3.120402827218362, −2.621326373857771, −1.687896780576015, −1.196731738688019, −0.3858991303882855, 0.3858991303882855, 1.196731738688019, 1.687896780576015, 2.621326373857771, 3.120402827218362, 4.269355287127342, 4.355649316657311, 5.029127772418323, 5.512175547564607, 5.788059683546901, 6.693602839251429, 7.094075861410001, 7.655106326071043, 8.015139192768306, 8.640400140832530, 9.163377049416901, 9.836634857209514, 10.32361441493818, 10.83984637092712, 11.32403705532243, 11.66169848830913, 12.00228016720157, 12.32089961728670, 12.99627605409502, 13.72582329503388

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