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Properties

Label 2-57960-1.1-c1-0-14
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 − 11-s + 5·13-s + 7·17-s − 4·19-s − 23-s + 25-s + 9·29-s + 6·31-s + 35-s − 4·37-s − 8·41-s − 2·43-s + 5·47-s + 49-s − 6·53-s + 55-s − 6·59-s + 10·61-s − 5·65-s + 16·67-s − 10·73-s + 77-s − 79-s + 4·83-s − 7·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.301·11-s + 1.38·13-s + 1.69·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 1.67·29-s + 1.07·31-s + 0.169·35-s − 0.657·37-s − 1.24·41-s − 0.304·43-s + 0.729·47-s + 1/7·49-s − 0.824·53-s + 0.134·55-s − 0.781·59-s + 1.28·61-s − 0.620·65-s + 1.95·67-s − 1.17·73-s + 0.113·77-s − 0.112·79-s + 0.439·83-s − 0.759·85-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

\(L(1)\) \(\approx\) \(2.373004609\)
\(L(\frac12)\) \(\approx\) \(2.373004609\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−14.19595692879708, −13.95486830505030, −13.30437571434129, −12.85967811613699, −12.18607199517590, −11.97298811663638, −11.34827399872575, −10.63725634926120, −10.30230342947856, −9.912637530915271, −9.111850561778533, −8.490448652478448, −8.183745851264878, −7.753705311792590, −6.800566252416755, −6.548722505384457, −5.905905830851408, −5.311477302967003, −4.653567872083508, −3.998252120797470, −3.362694314687770, −3.024617098786214, −2.076739395488319, −1.201482813178088, −0.5934233535368408, 0.5934233535368408, 1.201482813178088, 2.076739395488319, 3.024617098786214, 3.362694314687770, 3.998252120797470, 4.653567872083508, 5.311477302967003, 5.905905830851408, 6.548722505384457, 6.800566252416755, 7.753705311792590, 8.183745851264878, 8.490448652478448, 9.111850561778533, 9.912637530915271, 10.30230342947856, 10.63725634926120, 11.34827399872575, 11.97298811663638, 12.18607199517590, 12.85967811613699, 13.30437571434129, 13.95486830505030, 14.19595692879708

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