Properties

Label 2-2960-1.1-c1-0-34
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s − 2·7-s + 9-s + 2·13-s + 2·15-s + 6·17-s − 2·19-s − 4·21-s + 25-s − 4·27-s + 6·29-s + 10·31-s − 2·35-s + 37-s + 4·39-s − 6·41-s + 4·43-s + 45-s + 6·47-s − 3·49-s + 12·51-s + 6·53-s − 4·57-s + 6·59-s − 10·61-s − 2·63-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.458·19-s − 0.872·21-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 1.79·31-s − 0.338·35-s + 0.164·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s − 0.529·57-s + 0.781·59-s − 1.28·61-s − 0.251·63-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.957086323\)
\(L(\frac12)\) \(\approx\) \(2.957086323\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−8.664198216115629432922194533208, −8.191103303293285704368637540717, −7.38307853599763365662944352653, −6.40980423427948262053439128925, −5.87742893251240658386511463008, −4.78046421920466718576600337385, −3.69764756556049564109121180323, −3.06956943298945243015211313058, −2.32883355551645860271045039470, −1.03875136195910076855341646601, 1.03875136195910076855341646601, 2.32883355551645860271045039470, 3.06956943298945243015211313058, 3.69764756556049564109121180323, 4.78046421920466718576600337385, 5.87742893251240658386511463008, 6.40980423427948262053439128925, 7.38307853599763365662944352653, 8.191103303293285704368637540717, 8.664198216115629432922194533208

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