Properties

Label 2-6762-1.1-c1-0-75
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 + 1.56·5-s − 6-s − 8-s + 9-s − 1.56·10-s + 5.12·11-s + 12-s + 3.56·13-s + 1.56·15-s + 16-s + 1.12·17-s − 18-s + 5.12·19-s + 1.56·20-s − 5.12·22-s + 23-s − 24-s − 2.56·25-s − 3.56·26-s + 27-s + 7.56·29-s − 1.56·30-s − 3.12·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.698·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.493·10-s + 1.54·11-s + 0.288·12-s + 0.987·13-s + 0.403·15-s + 0.250·16-s + 0.272·17-s − 0.235·18-s + 1.17·19-s + 0.349·20-s − 1.09·22-s + 0.208·23-s − 0.204·24-s − 0.512·25-s − 0.698·26-s + 0.192·27-s + 1.40·29-s − 0.285·30-s − 0.560·31-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.797718466\)
\(L(\frac12)\) \(\approx\) \(2.797718466\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56T + 5T^{2} \)
11 \( 1 - 5.12T + 11T^{2} \)
13 \( 1 - 3.56T + 13T^{2} \)
17 \( 1 - 1.12T + 17T^{2} \)
19 \( 1 - 5.12T + 19T^{2} \)
29 \( 1 - 7.56T + 29T^{2} \)
31 \( 1 + 3.12T + 31T^{2} \)
37 \( 1 + 1.56T + 37T^{2} \)
41 \( 1 + 3.56T + 41T^{2} \)
43 \( 1 - 6.68T + 43T^{2} \)
47 \( 1 + 2.43T + 47T^{2} \)
53 \( 1 - 14.2T + 53T^{2} \)
59 \( 1 + 4.87T + 59T^{2} \)
61 \( 1 + 0.876T + 61T^{2} \)
67 \( 1 + 1.12T + 67T^{2} \)
71 \( 1 + 9.36T + 71T^{2} \)
73 \( 1 + 9.12T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−8.098529488235895040804031451489, −7.33246868998263460147698871430, −6.65300365337857999996177547636, −6.07014854691518564683477154056, −5.31694241620729611182723958908, −4.14162751096150190235662689761, −3.48889131846996885650509739754, −2.63205527387502566004539486049, −1.55854548603685245028628870858, −1.06835356316311672740392011791, 1.06835356316311672740392011791, 1.55854548603685245028628870858, 2.63205527387502566004539486049, 3.48889131846996885650509739754, 4.14162751096150190235662689761, 5.31694241620729611182723958908, 6.07014854691518564683477154056, 6.65300365337857999996177547636, 7.33246868998263460147698871430, 8.098529488235895040804031451489

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