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Properties

Label 2-57960-1.1-c1-0-15
Degree $2$
Conductor $57960$
Sign $1$
Analytic cond. $462.812$
Root an. cond. $21.5130$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s + 2·11-s + 6·17-s + 4·19-s − 23-s + 25-s − 2·29-s + 2·31-s − 35-s + 2·41-s + 4·43-s + 4·47-s + 49-s + 2·55-s − 6·59-s + 6·61-s + 8·67-s − 12·71-s − 16·73-s − 2·77-s − 14·79-s + 4·83-s + 6·85-s + 6·89-s + 4·95-s + 18·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.603·11-s + 1.45·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s + 0.359·31-s − 0.169·35-s + 0.312·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.269·55-s − 0.781·59-s + 0.768·61-s + 0.977·67-s − 1.42·71-s − 1.87·73-s − 0.227·77-s − 1.57·79-s + 0.439·83-s + 0.650·85-s + 0.635·89-s + 0.410·95-s + 1.82·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(462.812\)
Root analytic conductor: \(21.5130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 57960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.150597053\)
\(L(\frac12)\) \(\approx\) \(3.150597053\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.21677580418001, −14.09591407025767, −13.15950848954679, −13.03627393180461, −12.19545930554382, −11.89974963554055, −11.43598840185034, −10.65965494900259, −10.12941337067574, −9.827446840168374, −9.166295093104492, −8.864696599036826, −8.025335695672918, −7.514995038145346, −7.086367595721053, −6.347722917998604, −5.777804792319488, −5.511038812639304, −4.663835651763352, −4.056757398178953, −3.309450453819285, −2.946276335240504, −2.034556020348658, −1.294740850234239, −0.6605316640742303, 0.6605316640742303, 1.294740850234239, 2.034556020348658, 2.946276335240504, 3.309450453819285, 4.056757398178953, 4.663835651763352, 5.511038812639304, 5.777804792319488, 6.347722917998604, 7.086367595721053, 7.514995038145346, 8.025335695672918, 8.864696599036826, 9.166295093104492, 9.827446840168374, 10.12941337067574, 10.65965494900259, 11.43598840185034, 11.89974963554055, 12.19545930554382, 13.03627393180461, 13.15950848954679, 14.09591407025767, 14.21677580418001

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