Properties

Label 2-6762-1.1-c1-0-10
Degree $2$
Conductor $6762$
Sign $1$
Analytic cond. $53.9948$
Root an. cond. $7.34811$
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  + 2-s − 3-s + 4-s − 1.23·5-s − 6-s + 8-s + 9-s − 1.23·10-s − 5.23·11-s − 12-s − 4.47·13-s + 1.23·15-s + 16-s + 4·17-s + 18-s − 5.70·19-s − 1.23·20-s − 5.23·22-s + 23-s − 24-s − 3.47·25-s − 4.47·26-s − 27-s − 4.47·29-s + 1.23·30-s + 2.47·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.552·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.390·10-s − 1.57·11-s − 0.288·12-s − 1.24·13-s + 0.319·15-s + 0.250·16-s + 0.970·17-s + 0.235·18-s − 1.30·19-s − 0.276·20-s − 1.11·22-s + 0.208·23-s − 0.204·24-s − 0.694·25-s − 0.877·26-s − 0.192·27-s − 0.830·29-s + 0.225·30-s + 0.444·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6762\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(53.9948\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.355011501\)
\(L(\frac12)\) \(\approx\) \(1.355011501\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + 1.23T + 5T^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 5.70T + 19T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 5.23T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 + 0.763T + 61T^{2} \)
67 \( 1 - 9.70T + 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 - 4.47T + 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 0.472T + 97T^{2} \)
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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

−7.85176760319388738683138933320, −7.28223723235191053100342360186, −6.49283624411354344663123252064, −5.65035145052256515520646537374, −5.16710467432510798123569150681, −4.48809975862707633011525351641, −3.75225692261898516959088259902, −2.73166704462028131537592896456, −2.09816380183877348173326297853, −0.52016496873577664065999582433, 0.52016496873577664065999582433, 2.09816380183877348173326297853, 2.73166704462028131537592896456, 3.75225692261898516959088259902, 4.48809975862707633011525351641, 5.16710467432510798123569150681, 5.65035145052256515520646537374, 6.49283624411354344663123252064, 7.28223723235191053100342360186, 7.85176760319388738683138933320

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