Properties

Label 2-6762-1.1-c1-0-105
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 − 5-s − 6-s − 8-s + 9-s + 10-s − 3.65·11-s + 12-s + 0.851·13-s − 15-s + 16-s + 6.16·17-s − 18-s − 5.50·19-s − 20-s + 3.65·22-s − 23-s − 24-s − 4·25-s − 0.851·26-s + 27-s + 4.14·29-s + 30-s − 3.35·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.10·11-s + 0.288·12-s + 0.236·13-s − 0.258·15-s + 0.250·16-s + 1.49·17-s − 0.235·18-s − 1.26·19-s − 0.223·20-s + 0.779·22-s − 0.208·23-s − 0.204·24-s − 0.800·25-s − 0.167·26-s + 0.192·27-s + 0.770·29-s + 0.182·30-s − 0.603·31-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 \( 1 + T + 5T^{2} \)
11 \( 1 + 3.65T + 11T^{2} \)
13 \( 1 - 0.851T + 13T^{2} \)
17 \( 1 - 6.16T + 17T^{2} \)
19 \( 1 + 5.50T + 19T^{2} \)
29 \( 1 - 4.14T + 29T^{2} \)
31 \( 1 + 3.35T + 31T^{2} \)
37 \( 1 - 6.81T + 37T^{2} \)
41 \( 1 - 2.49T + 41T^{2} \)
43 \( 1 - 7.30T + 43T^{2} \)
47 \( 1 + 2.35T + 47T^{2} \)
53 \( 1 - 6.30T + 53T^{2} \)
59 \( 1 + 3.16T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 + 8.45T + 71T^{2} \)
73 \( 1 + 1.34T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 - 6.81T + 89T^{2} \)
97 \( 1 + 0.444T + 97T^{2} \)
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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

−7.68567295364639754076261775126, −7.39180015764187245219969529913, −6.22401075285146325147018363873, −5.71884888552695080086015297105, −4.64147146246157067857948223365, −3.86663962663737701080918623063, −2.98837117641336030369224435805, −2.32172878093346651093307815324, −1.24479070695563246709164466722, 0, 1.24479070695563246709164466722, 2.32172878093346651093307815324, 2.98837117641336030369224435805, 3.86663962663737701080918623063, 4.64147146246157067857948223365, 5.71884888552695080086015297105, 6.22401075285146325147018363873, 7.39180015764187245219969529913, 7.68567295364639754076261775126

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