Properties

Label 2-6762-1.1-c1-0-72
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 1.82·5-s + 6-s + 8-s + 9-s + 1.82·10-s + 4.14·11-s + 12-s − 2.18·13-s + 1.82·15-s + 16-s − 3.35·17-s + 18-s − 7.33·19-s + 1.82·20-s + 4.14·22-s + 23-s + 24-s − 1.65·25-s − 2.18·26-s + 27-s + 5.18·29-s + 1.82·30-s + 10.5·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.817·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.578·10-s + 1.25·11-s + 0.288·12-s − 0.606·13-s + 0.472·15-s + 0.250·16-s − 0.814·17-s + 0.235·18-s − 1.68·19-s + 0.408·20-s + 0.884·22-s + 0.208·23-s + 0.204·24-s − 0.331·25-s − 0.428·26-s + 0.192·27-s + 0.962·29-s + 0.333·30-s + 1.88·31-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

\(L(1)\) \(\approx\) \(5.116890793\)
\(L(\frac12)\) \(\approx\) \(5.116890793\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.82T + 5T^{2} \)
11 \( 1 - 4.14T + 11T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 + 3.35T + 17T^{2} \)
19 \( 1 + 7.33T + 19T^{2} \)
29 \( 1 - 5.18T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + 1.03T + 37T^{2} \)
41 \( 1 - 7.33T + 41T^{2} \)
43 \( 1 - 5.36T + 43T^{2} \)
47 \( 1 - 7.22T + 47T^{2} \)
53 \( 1 - 4.54T + 53T^{2} \)
59 \( 1 + 0.471T + 59T^{2} \)
61 \( 1 + 14.7T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 5.18T + 79T^{2} \)
83 \( 1 - 5.80T + 83T^{2} \)
89 \( 1 - 6.69T + 89T^{2} \)
97 \( 1 + 4.47T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.991675350414034688824779974272, −7.05791374646587897134608908557, −6.34985387798646355541105006991, −6.15689054576363608275828189829, −4.93033541121746925577249974103, −4.34902079629440952627410384293, −3.74412904500221142071997334264, −2.46588704901395158947411489153, −2.27409989565074935112491614219, −1.05718014400688246872445867769, 1.05718014400688246872445867769, 2.27409989565074935112491614219, 2.46588704901395158947411489153, 3.74412904500221142071997334264, 4.34902079629440952627410384293, 4.93033541121746925577249974103, 6.15689054576363608275828189829, 6.34985387798646355541105006991, 7.05791374646587897134608908557, 7.991675350414034688824779974272

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