Properties

Label 2-2960-1.1-c1-0-60
Degree $2$
Conductor $2960$
Sign $-1$
Analytic cond. $23.6357$
Root an. cond. $4.86165$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.363·3-s + 5-s − 0.363·7-s − 2.86·9-s + 1.14·11-s + 0.363·15-s − 4.77·19-s − 0.132·21-s − 4·23-s + 25-s − 2.13·27-s + 4.28·29-s − 5.50·31-s + 0.414·33-s − 0.363·35-s + 37-s + 3.86·41-s + 2.28·43-s − 2.86·45-s − 12.9·47-s − 6.86·49-s − 6.15·53-s + 1.14·55-s − 1.73·57-s − 7.78·59-s − 0.546·61-s + 1.04·63-s + ⋯
L(s)  = 1  + 0.209·3-s + 0.447·5-s − 0.137·7-s − 0.955·9-s + 0.344·11-s + 0.0938·15-s − 1.09·19-s − 0.0288·21-s − 0.834·23-s + 0.200·25-s − 0.410·27-s + 0.795·29-s − 0.988·31-s + 0.0721·33-s − 0.0614·35-s + 0.164·37-s + 0.604·41-s + 0.348·43-s − 0.427·45-s − 1.88·47-s − 0.981·49-s − 0.844·53-s + 0.153·55-s − 0.229·57-s − 1.01·59-s − 0.0699·61-s + 0.131·63-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: no
Arithmetic: yes
Character: $\chi_{2960} (1, \cdot )$
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 \( 1 - 0.363T + 3T^{2} \)
7 \( 1 + 0.363T + 7T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 4.77T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 4.28T + 29T^{2} \)
31 \( 1 + 5.50T + 31T^{2} \)
41 \( 1 - 3.86T + 41T^{2} \)
43 \( 1 - 2.28T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 6.15T + 53T^{2} \)
59 \( 1 + 7.78T + 59T^{2} \)
61 \( 1 + 0.546T + 61T^{2} \)
67 \( 1 + 0.212T + 67T^{2} \)
71 \( 1 + 3.58T + 71T^{2} \)
73 \( 1 - 8.69T + 73T^{2} \)
79 \( 1 - 7.06T + 79T^{2} \)
83 \( 1 - 0.466T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−8.337289579301630461961522488422, −7.83799449061939219747535230609, −6.61555926967380592267472205265, −6.21051063510896026392878197772, −5.35341608069528347382955764945, −4.43683025583940637510222960265, −3.47414873102356570411570152867, −2.58805865444571387122063407772, −1.65187748319839035007836031412, 0, 1.65187748319839035007836031412, 2.58805865444571387122063407772, 3.47414873102356570411570152867, 4.43683025583940637510222960265, 5.35341608069528347382955764945, 6.21051063510896026392878197772, 6.61555926967380592267472205265, 7.83799449061939219747535230609, 8.337289579301630461961522488422

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