Properties

Label 2-2960-1.1-c1-0-64
Degree $2$
Conductor $2960$
Sign $-1$
Analytic cond. $23.6357$
Root an. cond. $4.86165$
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  + 3-s + 5-s − 7-s − 2·9-s + 3·11-s − 6·13-s + 15-s − 21-s − 2·23-s + 25-s − 5·27-s − 6·29-s + 3·33-s − 35-s − 37-s − 6·39-s − 9·41-s + 10·43-s − 2·45-s − 47-s − 6·49-s + 53-s + 3·55-s − 12·61-s + 2·63-s − 6·65-s − 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 1.66·13-s + 0.258·15-s − 0.218·21-s − 0.417·23-s + 1/5·25-s − 0.962·27-s − 1.11·29-s + 0.522·33-s − 0.169·35-s − 0.164·37-s − 0.960·39-s − 1.40·41-s + 1.52·43-s − 0.298·45-s − 0.145·47-s − 6/7·49-s + 0.137·53-s + 0.404·55-s − 1.53·61-s + 0.251·63-s − 0.744·65-s − 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(23.6357\)
Root analytic conductor: \(4.86165\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2960,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
37 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.454669908512517207034899469370, −7.60623153841211391363563986699, −6.94299637524333632172982066183, −6.06920600436887629708345648593, −5.35180875187622235918959951272, −4.38768963429872255146818666900, −3.41377076281611111322968390654, −2.60923523958215693481233001490, −1.74532098704194312687928737335, 0, 1.74532098704194312687928737335, 2.60923523958215693481233001490, 3.41377076281611111322968390654, 4.38768963429872255146818666900, 5.35180875187622235918959951272, 6.06920600436887629708345648593, 6.94299637524333632172982066183, 7.60623153841211391363563986699, 8.454669908512517207034899469370

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