Properties

Label 2-6762-1.1-c1-0-122
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 − 0.722·5-s − 6-s + 8-s + 9-s − 0.722·10-s + 1.13·11-s − 12-s − 1.27·13-s + 0.722·15-s + 16-s + 3.93·17-s + 18-s − 2·19-s − 0.722·20-s + 1.13·22-s − 23-s − 24-s − 4.47·25-s − 1.27·26-s − 27-s − 5.76·29-s + 0.722·30-s − 6.20·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.323·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.228·10-s + 0.341·11-s − 0.288·12-s − 0.354·13-s + 0.186·15-s + 0.250·16-s + 0.954·17-s + 0.235·18-s − 0.458·19-s − 0.161·20-s + 0.241·22-s − 0.208·23-s − 0.204·24-s − 0.895·25-s − 0.250·26-s − 0.192·27-s − 1.07·29-s + 0.131·30-s − 1.11·31-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 \( 1 + 0.722T + 5T^{2} \)
11 \( 1 - 1.13T + 11T^{2} \)
13 \( 1 + 1.27T + 13T^{2} \)
17 \( 1 - 3.93T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 + 5.76T + 29T^{2} \)
31 \( 1 + 6.20T + 31T^{2} \)
37 \( 1 - 6.08T + 37T^{2} \)
41 \( 1 + 6.65T + 41T^{2} \)
43 \( 1 - 5.52T + 43T^{2} \)
47 \( 1 + 3.01T + 47T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 0.290T + 73T^{2} \)
79 \( 1 - 0.290T + 79T^{2} \)
83 \( 1 - 8.42T + 83T^{2} \)
89 \( 1 + 0.0631T + 89T^{2} \)
97 \( 1 + 0.750T + 97T^{2} \)
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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

−7.66294424305391222648562516571, −6.77658072219514281485973760402, −6.10105958852863133789494858518, −5.49917491340100099267445805055, −4.81552737757104401961765908020, −3.96149356923393788835711542590, −3.47297623075049083184693589518, −2.31497319533247266143115321025, −1.39341535780073484769676425262, 0, 1.39341535780073484769676425262, 2.31497319533247266143115321025, 3.47297623075049083184693589518, 3.96149356923393788835711542590, 4.81552737757104401961765908020, 5.49917491340100099267445805055, 6.10105958852863133789494858518, 6.77658072219514281485973760402, 7.66294424305391222648562516571

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