Properties

Label 2-7600-1.1-c1-0-105
Degree $2$
Conductor $7600$
Sign $1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
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  + 1.53·3-s + 5.03·7-s − 0.633·9-s + 3.03·11-s + 4.57·13-s + 1.07·17-s − 19-s + 7.74·21-s + 4.11·23-s − 5.58·27-s − 1.07·29-s − 5.58·31-s + 4.66·33-s − 0.0947·37-s + 7.03·39-s + 10.6·41-s + 5.03·43-s − 12.2·47-s + 18.3·49-s + 1.65·51-s + 4.09·53-s − 1.53·57-s + 1.39·59-s − 5.69·61-s − 3.18·63-s + 5.28·67-s + 6.32·69-s + ⋯
L(s)  = 1  + 0.888·3-s + 1.90·7-s − 0.211·9-s + 0.914·11-s + 1.26·13-s + 0.261·17-s − 0.229·19-s + 1.68·21-s + 0.857·23-s − 1.07·27-s − 0.199·29-s − 1.00·31-s + 0.812·33-s − 0.0155·37-s + 1.12·39-s + 1.66·41-s + 0.767·43-s − 1.79·47-s + 2.61·49-s + 0.231·51-s + 0.562·53-s − 0.203·57-s + 0.182·59-s − 0.729·61-s − 0.401·63-s + 0.645·67-s + 0.761·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7600\)    =    \(2^{4} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(60.6863\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.235956293\)
\(L(\frac12)\) \(\approx\) \(4.235956293\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 1.53T + 3T^{2} \)
7 \( 1 - 5.03T + 7T^{2} \)
11 \( 1 - 3.03T + 11T^{2} \)
13 \( 1 - 4.57T + 13T^{2} \)
17 \( 1 - 1.07T + 17T^{2} \)
23 \( 1 - 4.11T + 23T^{2} \)
29 \( 1 + 1.07T + 29T^{2} \)
31 \( 1 + 5.58T + 31T^{2} \)
37 \( 1 + 0.0947T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 5.03T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 - 4.09T + 53T^{2} \)
59 \( 1 - 1.39T + 59T^{2} \)
61 \( 1 + 5.69T + 61T^{2} \)
67 \( 1 - 5.28T + 67T^{2} \)
71 \( 1 - 5.67T + 71T^{2} \)
73 \( 1 + 9.07T + 73T^{2} \)
79 \( 1 - 5.39T + 79T^{2} \)
83 \( 1 - 1.95T + 83T^{2} \)
89 \( 1 + 2.18T + 89T^{2} \)
97 \( 1 - 2.16T + 97T^{2} \)
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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

−7.88067950627968385321727626750, −7.53254155431020257146628319848, −6.51105257898596533211624259478, −5.71510684513258739717011804516, −5.04431333864212017147356434928, −4.12967513917981083346706199943, −3.65578815590354439471200081997, −2.62264255388332758007980102369, −1.73545798396630404152577041266, −1.11370411177691766805520463828, 1.11370411177691766805520463828, 1.73545798396630404152577041266, 2.62264255388332758007980102369, 3.65578815590354439471200081997, 4.12967513917981083346706199943, 5.04431333864212017147356434928, 5.71510684513258739717011804516, 6.51105257898596533211624259478, 7.53254155431020257146628319848, 7.88067950627968385321727626750

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