Properties

Label 2-3017-1.1-c1-0-5
Degree $2$
Conductor $3017$
Sign $1$
Analytic cond. $24.0908$
Root an. cond. $4.90824$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.07·2-s − 2.49·3-s − 0.843·4-s − 0.878·5-s − 2.68·6-s − 7-s − 3.05·8-s + 3.23·9-s − 0.945·10-s − 1.67·11-s + 2.10·12-s − 4.09·13-s − 1.07·14-s + 2.19·15-s − 1.60·16-s − 2.80·17-s + 3.48·18-s + 2.51·19-s + 0.741·20-s + 2.49·21-s − 1.80·22-s − 6.81·23-s + 7.63·24-s − 4.22·25-s − 4.40·26-s − 0.595·27-s + 0.843·28-s + ⋯
L(s)  = 1  + 0.760·2-s − 1.44·3-s − 0.421·4-s − 0.393·5-s − 1.09·6-s − 0.377·7-s − 1.08·8-s + 1.07·9-s − 0.298·10-s − 0.505·11-s + 0.607·12-s − 1.13·13-s − 0.287·14-s + 0.566·15-s − 0.400·16-s − 0.679·17-s + 0.821·18-s + 0.577·19-s + 0.165·20-s + 0.545·21-s − 0.384·22-s − 1.42·23-s + 1.55·24-s − 0.845·25-s − 0.863·26-s − 0.114·27-s + 0.159·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3017\)    =    \(7 \cdot 431\)
Sign: $1$
Analytic conductor: \(24.0908\)
Root analytic conductor: \(4.90824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3017,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1505811505\)
\(L(\frac12)\) \(\approx\) \(0.1505811505\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
431 \( 1 - T \)
good2 \( 1 - 1.07T + 2T^{2} \)
3 \( 1 + 2.49T + 3T^{2} \)
5 \( 1 + 0.878T + 5T^{2} \)
11 \( 1 + 1.67T + 11T^{2} \)
13 \( 1 + 4.09T + 13T^{2} \)
17 \( 1 + 2.80T + 17T^{2} \)
19 \( 1 - 2.51T + 19T^{2} \)
23 \( 1 + 6.81T + 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + 9.36T + 31T^{2} \)
37 \( 1 + 1.74T + 37T^{2} \)
41 \( 1 + 2.97T + 41T^{2} \)
43 \( 1 + 1.34T + 43T^{2} \)
47 \( 1 - 6.62T + 47T^{2} \)
53 \( 1 + 1.41T + 53T^{2} \)
59 \( 1 + 2.84T + 59T^{2} \)
61 \( 1 - 9.36T + 61T^{2} \)
67 \( 1 + 4.52T + 67T^{2} \)
71 \( 1 + 8.93T + 71T^{2} \)
73 \( 1 + 1.43T + 73T^{2} \)
79 \( 1 + 3.05T + 79T^{2} \)
83 \( 1 + 1.20T + 83T^{2} \)
89 \( 1 - 7.83T + 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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

−8.822229220352224897315132714685, −7.68791937307479355974373342200, −7.09969653223611875539472278175, −6.06116855450410784685198753194, −5.61036017253547364190308070224, −4.97352787713671535244327985555, −4.20968711324800175803245887258, −3.44967462239677289706449029397, −2.14042188737499373803992884878, −0.21260804029347922987373417653, 0.21260804029347922987373417653, 2.14042188737499373803992884878, 3.44967462239677289706449029397, 4.20968711324800175803245887258, 4.97352787713671535244327985555, 5.61036017253547364190308070224, 6.06116855450410784685198753194, 7.09969653223611875539472278175, 7.68791937307479355974373342200, 8.822229220352224897315132714685

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