Properties

Degree $2$
Conductor $6762$
Sign $1$
Motivic weight $1$
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 + 2·5-s + 6-s + 8-s + 9-s + 2·10-s − 4·11-s + 12-s + 2·13-s + 2·15-s + 16-s + 6·17-s + 18-s − 4·19-s + 2·20-s − 4·22-s − 23-s + 24-s − 25-s + 2·26-s + 27-s − 2·29-s + 2·30-s + 8·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.852·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.371·29-s + 0.365·30-s + 1.43·31-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: $1$
Motivic weight: \(1\)
Character: $\chi_{6762} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.005206899\)
\(L(\frac12)\) \(\approx\) \(5.005206899\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−17.12060055650168, −16.59044315560334, −15.93793279354187, −15.34184246823682, −14.83192424392753, −14.12800963163420, −13.71539573762196, −13.16929528400606, −12.70638360479940, −12.03833416149722, −11.21085719196195, −10.43226559698012, −10.07757697417951, −9.430010139823967, −8.475928535774986, −7.953818650031663, −7.329644117537405, −6.330915389656806, −5.834979220891873, −5.221241566744915, −4.342571722960689, −3.560914615569478, −2.660222951512286, −2.179331016859398, −1.040732004656004, 1.040732004656004, 2.179331016859398, 2.660222951512286, 3.560914615569478, 4.342571722960689, 5.221241566744915, 5.834979220891873, 6.330915389656806, 7.329644117537405, 7.953818650031663, 8.475928535774986, 9.430010139823967, 10.07757697417951, 10.43226559698012, 11.21085719196195, 12.03833416149722, 12.70638360479940, 13.16929528400606, 13.71539573762196, 14.12800963163420, 14.83192424392753, 15.34184246823682, 15.93793279354187, 16.59044315560334, 17.12060055650168

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