Properties

Label 2-7600-1.1-c1-0-63
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  − 3.03·3-s + 2.46·7-s + 6.19·9-s − 0.728·11-s + 6.23·13-s − 0.563·17-s + 19-s − 7.49·21-s + 4.63·23-s − 9.70·27-s + 10.2·29-s − 6.06·31-s + 2.21·33-s + 5.72·37-s − 18.9·39-s + 4.79·41-s + 8.06·43-s + 8.12·47-s − 0.900·49-s + 1.70·51-s + 1.53·53-s − 3.03·57-s + 5.76·59-s + 10.9·61-s + 15.3·63-s + 12.9·67-s − 14.0·69-s + ⋯
L(s)  = 1  − 1.75·3-s + 0.933·7-s + 2.06·9-s − 0.219·11-s + 1.72·13-s − 0.136·17-s + 0.229·19-s − 1.63·21-s + 0.966·23-s − 1.86·27-s + 1.89·29-s − 1.08·31-s + 0.384·33-s + 0.941·37-s − 3.02·39-s + 0.748·41-s + 1.23·43-s + 1.18·47-s − 0.128·49-s + 0.239·51-s + 0.210·53-s − 0.401·57-s + 0.750·59-s + 1.40·61-s + 1.92·63-s + 1.58·67-s − 1.69·69-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

\(L(1)\) \(\approx\) \(1.592482773\)
\(L(\frac12)\) \(\approx\) \(1.592482773\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.03T + 3T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
11 \( 1 + 0.728T + 11T^{2} \)
13 \( 1 - 6.23T + 13T^{2} \)
17 \( 1 + 0.563T + 17T^{2} \)
23 \( 1 - 4.63T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 + 6.06T + 31T^{2} \)
37 \( 1 - 5.72T + 37T^{2} \)
41 \( 1 - 4.79T + 41T^{2} \)
43 \( 1 - 8.06T + 43T^{2} \)
47 \( 1 - 8.12T + 47T^{2} \)
53 \( 1 - 1.53T + 53T^{2} \)
59 \( 1 - 5.76T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 4.39T + 71T^{2} \)
73 \( 1 - 4.09T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 - 7.85T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−7.82142753175158629334238182109, −6.89367407945250005510228424173, −6.49021737792074664408740277859, −5.52884321930316538402834562022, −5.40480110732223830788625411585, −4.40094898341595333935509734015, −3.93192501361106391519041576464, −2.57893078085054921112968065139, −1.27018085369507755373299784058, −0.845526121389797306699601912610, 0.845526121389797306699601912610, 1.27018085369507755373299784058, 2.57893078085054921112968065139, 3.93192501361106391519041576464, 4.40094898341595333935509734015, 5.40480110732223830788625411585, 5.52884321930316538402834562022, 6.49021737792074664408740277859, 6.89367407945250005510228424173, 7.82142753175158629334238182109

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