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Properties

Label 2-57960-1.1-c1-0-46
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s + 2·13-s + 2·17-s + 4·19-s − 23-s + 25-s + 2·29-s − 35-s + 2·37-s − 2·41-s + 4·43-s − 12·47-s + 49-s − 2·53-s + 12·59-s − 6·61-s + 2·65-s − 4·67-s − 10·73-s − 4·79-s + 12·83-s + 2·85-s − 6·89-s − 2·91-s + 4·95-s − 2·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 0.169·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.274·53-s + 1.56·59-s − 0.768·61-s + 0.248·65-s − 0.488·67-s − 1.17·73-s − 0.450·79-s + 1.31·83-s + 0.216·85-s − 0.635·89-s − 0.209·91-s + 0.410·95-s − 0.203·97-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: \(1\)
Selberg data: \((2,\ 57960,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−14.68945694163324, −13.97005083807205, −13.64249416821151, −13.09832847447472, −12.70528628120042, −11.95406350942393, −11.69380069211918, −11.00519380246918, −10.43576286724784, −10.00731109589818, −9.422088153499728, −9.114648358070461, −8.274323030969722, −7.961853634344396, −7.228902775369532, −6.651051882603810, −6.196648123417816, −5.526945461042974, −5.160687854774856, −4.331554098823921, −3.722805297652374, −3.058915470243689, −2.572552664940027, −1.580918244975788, −1.084486144647524, 0, 1.084486144647524, 1.580918244975788, 2.572552664940027, 3.058915470243689, 3.722805297652374, 4.331554098823921, 5.160687854774856, 5.526945461042974, 6.196648123417816, 6.651051882603810, 7.228902775369532, 7.961853634344396, 8.274323030969722, 9.114648358070461, 9.422088153499728, 10.00731109589818, 10.43576286724784, 11.00519380246918, 11.69380069211918, 11.95406350942393, 12.70528628120042, 13.09832847447472, 13.64249416821151, 13.97005083807205, 14.68945694163324

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