Properties

Label 2-6762-1.1-c1-0-121
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 1.49·5-s − 6-s + 8-s + 9-s − 1.49·10-s + 5.90·11-s − 12-s − 0.505·13-s + 1.49·15-s + 16-s − 6.55·17-s + 18-s − 2·19-s − 1.49·20-s + 5.90·22-s − 23-s − 24-s − 2.76·25-s − 0.505·26-s − 27-s + 7.03·29-s + 1.49·30-s − 5.26·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.668·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.472·10-s + 1.78·11-s − 0.288·12-s − 0.140·13-s + 0.385·15-s + 0.250·16-s − 1.58·17-s + 0.235·18-s − 0.458·19-s − 0.334·20-s + 1.25·22-s − 0.208·23-s − 0.204·24-s − 0.553·25-s − 0.0992·26-s − 0.192·27-s + 1.30·29-s + 0.272·30-s − 0.945·31-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: \(1\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.49T + 5T^{2} \)
11 \( 1 - 5.90T + 11T^{2} \)
13 \( 1 + 0.505T + 13T^{2} \)
17 \( 1 + 6.55T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 - 7.03T + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + 9.95T + 37T^{2} \)
41 \( 1 - 3.05T + 41T^{2} \)
43 \( 1 + 8.96T + 43T^{2} \)
47 \( 1 - 7.30T + 47T^{2} \)
53 \( 1 - 4.91T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 0.458T + 61T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 + 9.10T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 19.1T + 97T^{2} \)
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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

−7.17648853053727034501237473820, −6.90167084113978340443943711269, −6.26494607777064839359160874416, −5.50874753543384454169406578765, −4.53449067684653035382903269130, −4.13531101339035048897633406660, −3.51569374475604985635268536291, −2.28567286846969769122911738192, −1.37831191702156250099468176372, 0, 1.37831191702156250099468176372, 2.28567286846969769122911738192, 3.51569374475604985635268536291, 4.13531101339035048897633406660, 4.53449067684653035382903269130, 5.50874753543384454169406578765, 6.26494607777064839359160874416, 6.90167084113978340443943711269, 7.17648853053727034501237473820

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