Properties

Label 2-1305-1.1-c1-0-13
Degree $2$
Conductor $1305$
Sign $1$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
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  + 0.897·2-s − 1.19·4-s − 5-s + 3.83·7-s − 2.86·8-s − 0.897·10-s − 2.82·11-s + 0.989·13-s + 3.43·14-s − 0.183·16-s − 1.01·17-s + 5.13·19-s + 1.19·20-s − 2.53·22-s + 7.78·23-s + 25-s + 0.887·26-s − 4.57·28-s + 29-s − 1.13·31-s + 5.56·32-s − 0.907·34-s − 3.83·35-s + 10.7·37-s + 4.60·38-s + 2.86·40-s + 2.58·41-s + ⋯
L(s)  = 1  + 0.634·2-s − 0.597·4-s − 0.447·5-s + 1.44·7-s − 1.01·8-s − 0.283·10-s − 0.852·11-s + 0.274·13-s + 0.918·14-s − 0.0459·16-s − 0.245·17-s + 1.17·19-s + 0.267·20-s − 0.540·22-s + 1.62·23-s + 0.200·25-s + 0.174·26-s − 0.864·28-s + 0.185·29-s − 0.203·31-s + 0.984·32-s − 0.155·34-s − 0.647·35-s + 1.76·37-s + 0.747·38-s + 0.453·40-s + 0.403·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.971081491\)
\(L(\frac12)\) \(\approx\) \(1.971081491\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 - 0.897T + 2T^{2} \)
7 \( 1 - 3.83T + 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 - 0.989T + 13T^{2} \)
17 \( 1 + 1.01T + 17T^{2} \)
19 \( 1 - 5.13T + 19T^{2} \)
23 \( 1 - 7.78T + 23T^{2} \)
31 \( 1 + 1.13T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 2.58T + 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 - 0.252T + 47T^{2} \)
53 \( 1 - 5.70T + 53T^{2} \)
59 \( 1 + 3.66T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 3.75T + 67T^{2} \)
71 \( 1 + 6.26T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 1.06T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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

−9.503931397689155364165909571453, −8.769573984039085345139776523274, −7.978281630520333064167226453201, −7.40705690121689361384584776459, −6.07659878108074986668131091451, −4.97412404768892574795205244287, −4.86567948368613554644946815520, −3.67766041552856972600998491292, −2.65737882212744162530078298861, −0.991939427866663918337482067132, 0.991939427866663918337482067132, 2.65737882212744162530078298861, 3.67766041552856972600998491292, 4.86567948368613554644946815520, 4.97412404768892574795205244287, 6.07659878108074986668131091451, 7.40705690121689361384584776459, 7.978281630520333064167226453201, 8.769573984039085345139776523274, 9.503931397689155364165909571453

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