Properties

Label 2-6171-1.1-c1-0-135
Degree $2$
Conductor $6171$
Sign $1$
Analytic cond. $49.2756$
Root an. cond. $7.01966$
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  − 1.19·2-s + 3-s − 0.577·4-s − 0.343·5-s − 1.19·6-s + 4.47·7-s + 3.07·8-s + 9-s + 0.409·10-s − 0.577·12-s + 6.51·13-s − 5.34·14-s − 0.343·15-s − 2.51·16-s + 17-s − 1.19·18-s − 4.34·19-s + 0.198·20-s + 4.47·21-s + 1.90·23-s + 3.07·24-s − 4.88·25-s − 7.76·26-s + 27-s − 2.58·28-s + 3.02·29-s + 0.409·30-s + ⋯
L(s)  = 1  − 0.843·2-s + 0.577·3-s − 0.288·4-s − 0.153·5-s − 0.486·6-s + 1.69·7-s + 1.08·8-s + 0.333·9-s + 0.129·10-s − 0.166·12-s + 1.80·13-s − 1.42·14-s − 0.0886·15-s − 0.627·16-s + 0.242·17-s − 0.281·18-s − 0.997·19-s + 0.0443·20-s + 0.977·21-s + 0.396·23-s + 0.627·24-s − 0.976·25-s − 1.52·26-s + 0.192·27-s − 0.489·28-s + 0.562·29-s + 0.0747·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6171\)    =    \(3 \cdot 11^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(49.2756\)
Root analytic conductor: \(7.01966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6171,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.042205902\)
\(L(\frac12)\) \(\approx\) \(2.042205902\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 \)
17 \( 1 - T \)
good2 \( 1 + 1.19T + 2T^{2} \)
5 \( 1 + 0.343T + 5T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
13 \( 1 - 6.51T + 13T^{2} \)
19 \( 1 + 4.34T + 19T^{2} \)
23 \( 1 - 1.90T + 23T^{2} \)
29 \( 1 - 3.02T + 29T^{2} \)
31 \( 1 - 8.10T + 31T^{2} \)
37 \( 1 + 3.63T + 37T^{2} \)
41 \( 1 - 8.27T + 41T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 - 9.21T + 47T^{2} \)
53 \( 1 - 5.25T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 1.83T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + 3.65T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 - 5.73T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 - 9.45T + 97T^{2} \)
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

−8.167540005702027274546715845540, −7.80480125943611241576001972205, −6.92385449888611805291719421180, −5.94630007716656069713595531405, −5.08432246766956576365777524449, −4.21688138242437001717197705400, −3.91309002465402455621429122292, −2.50140805770798534946275035369, −1.54989771547452788989799421160, −0.946877909953613385830449182916, 0.946877909953613385830449182916, 1.54989771547452788989799421160, 2.50140805770798534946275035369, 3.91309002465402455621429122292, 4.21688138242437001717197705400, 5.08432246766956576365777524449, 5.94630007716656069713595531405, 6.92385449888611805291719421180, 7.80480125943611241576001972205, 8.167540005702027274546715845540

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