Properties

Label 2-7920-1.1-c1-0-1
Degree $2$
Conductor $7920$
Sign $1$
Analytic cond. $63.2415$
Root an. cond. $7.95245$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4.60·7-s + 11-s − 4.60·13-s − 6.60·17-s + 7.21·19-s + 25-s − 8·29-s − 9.21·31-s + 4.60·35-s − 3.21·37-s − 8·41-s + 3.39·43-s − 5.21·47-s + 14.2·49-s − 2·53-s − 55-s + 8·59-s + 7.21·61-s + 4.60·65-s + 4·67-s − 14.4·71-s + 0.605·73-s − 4.60·77-s + 11.2·79-s − 10.6·83-s + 6.60·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.74·7-s + 0.301·11-s − 1.27·13-s − 1.60·17-s + 1.65·19-s + 0.200·25-s − 1.48·29-s − 1.65·31-s + 0.778·35-s − 0.527·37-s − 1.24·41-s + 0.517·43-s − 0.760·47-s + 2.03·49-s − 0.274·53-s − 0.134·55-s + 1.04·59-s + 0.923·61-s + 0.571·65-s + 0.488·67-s − 1.71·71-s + 0.0708·73-s − 0.524·77-s + 1.26·79-s − 1.16·83-s + 0.716·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(63.2415\)
Root analytic conductor: \(7.95245\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4316592513\)
\(L(\frac12)\) \(\approx\) \(0.4316592513\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 - T \)
good7 \( 1 + 4.60T + 7T^{2} \)
13 \( 1 + 4.60T + 13T^{2} \)
17 \( 1 + 6.60T + 17T^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + 9.21T + 31T^{2} \)
37 \( 1 + 3.21T + 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 3.39T + 43T^{2} \)
47 \( 1 + 5.21T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 7.21T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 - 0.605T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−7.60017366118396307950429390639, −7.00891872598201563558799975826, −6.77102552405242496745151760111, −5.70158671667094594990176892892, −5.15385892152145313954897201307, −4.11232525833053061653749322358, −3.50025555899855534814454861781, −2.82983682538495676475230197901, −1.87555099932477660171282438902, −0.30564685641399426488259525472, 0.30564685641399426488259525472, 1.87555099932477660171282438902, 2.82983682538495676475230197901, 3.50025555899855534814454861781, 4.11232525833053061653749322358, 5.15385892152145313954897201307, 5.70158671667094594990176892892, 6.77102552405242496745151760111, 7.00891872598201563558799975826, 7.60017366118396307950429390639

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