Properties

Label 2-2960-1.1-c1-0-18
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.925·3-s − 5-s − 0.763·7-s − 2.14·9-s + 6.44·11-s − 2.67·13-s − 0.925·15-s + 5.30·17-s − 4.88·19-s − 0.706·21-s − 0.149·23-s + 25-s − 4.75·27-s + 5.21·29-s − 0.912·31-s + 5.96·33-s + 0.763·35-s + 37-s − 2.47·39-s + 10.7·41-s + 8.12·43-s + 2.14·45-s − 3.51·47-s − 6.41·49-s + 4.90·51-s + 10.5·53-s − 6.44·55-s + ⋯
L(s)  = 1  + 0.534·3-s − 0.447·5-s − 0.288·7-s − 0.714·9-s + 1.94·11-s − 0.740·13-s − 0.238·15-s + 1.28·17-s − 1.12·19-s − 0.154·21-s − 0.0312·23-s + 0.200·25-s − 0.915·27-s + 0.968·29-s − 0.163·31-s + 1.03·33-s + 0.129·35-s + 0.164·37-s − 0.395·39-s + 1.68·41-s + 1.23·43-s + 0.319·45-s − 0.511·47-s − 0.916·49-s + 0.686·51-s + 1.45·53-s − 0.869·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.981666360\)
\(L(\frac12)\) \(\approx\) \(1.981666360\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.925T + 3T^{2} \)
7 \( 1 + 0.763T + 7T^{2} \)
11 \( 1 - 6.44T + 11T^{2} \)
13 \( 1 + 2.67T + 13T^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
19 \( 1 + 4.88T + 19T^{2} \)
23 \( 1 + 0.149T + 23T^{2} \)
29 \( 1 - 5.21T + 29T^{2} \)
31 \( 1 + 0.912T + 31T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 8.12T + 43T^{2} \)
47 \( 1 + 3.51T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 4.45T + 59T^{2} \)
61 \( 1 + 8.26T + 61T^{2} \)
67 \( 1 - 6.88T + 67T^{2} \)
71 \( 1 + 2.03T + 71T^{2} \)
73 \( 1 - 9.81T + 73T^{2} \)
79 \( 1 - 1.83T + 79T^{2} \)
83 \( 1 + 1.03T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 + 5.18T + 97T^{2} \)
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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

−8.827127692710184844062684408548, −8.027362989072120879831714240246, −7.35286813041725819739773062643, −6.43536367349372381955790179986, −5.88367777555403592038435916499, −4.66868031112719792972498436426, −3.88891899918750285583614688970, −3.18938689142215102552038326759, −2.19343231803019547596089387265, −0.848165116974752923147731731111, 0.848165116974752923147731731111, 2.19343231803019547596089387265, 3.18938689142215102552038326759, 3.88891899918750285583614688970, 4.66868031112719792972498436426, 5.88367777555403592038435916499, 6.43536367349372381955790179986, 7.35286813041725819739773062643, 8.027362989072120879831714240246, 8.827127692710184844062684408548

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