Properties

Label 2-6762-1.1-c1-0-16
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 − 3.82·5-s + 6-s + 8-s + 9-s − 3.82·10-s − 5.10·11-s + 12-s + 0.0556·13-s − 3.82·15-s + 16-s − 6.77·17-s + 18-s + 4.16·19-s − 3.82·20-s − 5.10·22-s + 23-s + 24-s + 9.65·25-s + 0.0556·26-s + 27-s + 2.94·29-s − 3.82·30-s − 3.21·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.71·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.21·10-s − 1.54·11-s + 0.288·12-s + 0.0154·13-s − 0.988·15-s + 0.250·16-s − 1.64·17-s + 0.235·18-s + 0.955·19-s − 0.856·20-s − 1.08·22-s + 0.208·23-s + 0.204·24-s + 1.93·25-s + 0.0109·26-s + 0.192·27-s + 0.546·29-s − 0.698·30-s − 0.578·31-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: $\chi_{6762} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.053539719\)
\(L(\frac12)\) \(\approx\) \(2.053539719\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.82T + 5T^{2} \)
11 \( 1 + 5.10T + 11T^{2} \)
13 \( 1 - 0.0556T + 13T^{2} \)
17 \( 1 + 6.77T + 17T^{2} \)
19 \( 1 - 4.16T + 19T^{2} \)
29 \( 1 - 2.94T + 29T^{2} \)
31 \( 1 + 3.21T + 31T^{2} \)
37 \( 1 + 8.05T + 37T^{2} \)
41 \( 1 + 4.16T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 - 5.71T + 53T^{2} \)
59 \( 1 - 8.60T + 59T^{2} \)
61 \( 1 + 5.77T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 2.21T + 73T^{2} \)
79 \( 1 + 2.94T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 - 2.39T + 89T^{2} \)
97 \( 1 - 4.60T + 97T^{2} \)
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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

−7.87947267467851287355616755410, −7.20753035932398927074365845009, −6.94894663689195281880446370984, −5.64333648399509723370690061008, −4.95246240546459578889603385890, −4.26676312830454856589300227645, −3.67088411558871588578206117885, −2.88306889019999387035211139889, −2.23375557849041268040594889810, −0.61053783552254470587351278637, 0.61053783552254470587351278637, 2.23375557849041268040594889810, 2.88306889019999387035211139889, 3.67088411558871588578206117885, 4.26676312830454856589300227645, 4.95246240546459578889603385890, 5.64333648399509723370690061008, 6.94894663689195281880446370984, 7.20753035932398927074365845009, 7.87947267467851287355616755410

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