Properties

Label 2-6762-1.1-c1-0-27
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 − 3.22·5-s − 6-s + 8-s + 9-s − 3.22·10-s + 3.31·11-s − 12-s + 4.43·13-s + 3.22·15-s + 16-s + 0.0825·17-s + 18-s − 2·19-s − 3.22·20-s + 3.31·22-s − 23-s − 24-s + 5.43·25-s + 4.43·26-s − 27-s + 0.568·29-s + 3.22·30-s − 2·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.44·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.02·10-s + 0.998·11-s − 0.288·12-s + 1.22·13-s + 0.833·15-s + 0.250·16-s + 0.0200·17-s + 0.235·18-s − 0.458·19-s − 0.722·20-s + 0.706·22-s − 0.208·23-s − 0.204·24-s + 1.08·25-s + 0.869·26-s − 0.192·27-s + 0.105·29-s + 0.589·30-s − 0.359·31-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

\(L(1)\) \(\approx\) \(2.050501627\)
\(L(\frac12)\) \(\approx\) \(2.050501627\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.22T + 5T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
13 \( 1 - 4.43T + 13T^{2} \)
17 \( 1 - 0.0825T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 - 0.568T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2.45T + 37T^{2} \)
41 \( 1 - 8.45T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 7.62T + 47T^{2} \)
53 \( 1 - 14.0T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 4.86T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 7.69T + 71T^{2} \)
73 \( 1 + 3.86T + 73T^{2} \)
79 \( 1 - 4.68T + 79T^{2} \)
83 \( 1 + 0.165T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−7.75360782651541779600283666958, −7.24752321682157321666716359026, −6.36633802710694703440418008823, −6.03385429944781728595780405462, −4.98527839006043694776340462974, −4.19650946656072739250291855361, −3.86023863239725417501799202265, −3.11478002129955827300276283798, −1.73557922589564724389127673692, −0.70079929118970064296215145473, 0.70079929118970064296215145473, 1.73557922589564724389127673692, 3.11478002129955827300276283798, 3.86023863239725417501799202265, 4.19650946656072739250291855361, 4.98527839006043694776340462974, 6.03385429944781728595780405462, 6.36633802710694703440418008823, 7.24752321682157321666716359026, 7.75360782651541779600283666958

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