Properties

Label 2-770-1.1-c1-0-17
Degree $2$
Conductor $770$
Sign $1$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
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 + 2·3-s + 4-s + 5-s + 2·6-s + 7-s + 8-s + 9-s + 10-s − 11-s + 2·12-s + 6.74·13-s + 14-s + 2·15-s + 16-s − 6.74·17-s + 18-s − 6.74·19-s + 20-s + 2·21-s − 22-s − 6.74·23-s + 2·24-s + 25-s + 6.74·26-s − 4·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 0.5·4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s + 0.577·12-s + 1.87·13-s + 0.267·14-s + 0.516·15-s + 0.250·16-s − 1.63·17-s + 0.235·18-s − 1.54·19-s + 0.223·20-s + 0.436·21-s − 0.213·22-s − 1.40·23-s + 0.408·24-s + 0.200·25-s + 1.32·26-s − 0.769·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.615293320\)
\(L(\frac12)\) \(\approx\) \(3.615293320\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good3 \( 1 - 2T + 3T^{2} \)
13 \( 1 - 6.74T + 13T^{2} \)
17 \( 1 + 6.74T + 17T^{2} \)
19 \( 1 + 6.74T + 19T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
31 \( 1 - 4.74T + 31T^{2} \)
37 \( 1 - 0.744T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + 8.74T + 59T^{2} \)
61 \( 1 - 1.25T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 6.74T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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

−10.55348405566603840281525483099, −9.259740009163592005687544437399, −8.422076705731661692957613300515, −8.094738857572422564011297863893, −6.50910631703067432125052856275, −6.11423148148720500758859467564, −4.61521477311126278475137722645, −3.87372146782149442686012785408, −2.67990546288820200222337348371, −1.83323675261394832793354842974, 1.83323675261394832793354842974, 2.67990546288820200222337348371, 3.87372146782149442686012785408, 4.61521477311126278475137722645, 6.11423148148720500758859467564, 6.50910631703067432125052856275, 8.094738857572422564011297863893, 8.422076705731661692957613300515, 9.259740009163592005687544437399, 10.55348405566603840281525483099

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