Properties

Label 2-6015-1.1-c1-0-22
Degree $2$
Conductor $6015$
Sign $1$
Analytic cond. $48.0300$
Root an. cond. $6.93036$
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  − 0.347·2-s − 3-s − 1.87·4-s + 5-s + 0.347·6-s − 4.57·7-s + 1.34·8-s + 9-s − 0.347·10-s + 5.17·11-s + 1.87·12-s − 4.70·13-s + 1.59·14-s − 15-s + 3.28·16-s + 3.74·17-s − 0.347·18-s − 1.63·19-s − 1.87·20-s + 4.57·21-s − 1.79·22-s − 4.05·23-s − 1.34·24-s + 25-s + 1.63·26-s − 27-s + 8.60·28-s + ⋯
L(s)  = 1  − 0.245·2-s − 0.577·3-s − 0.939·4-s + 0.447·5-s + 0.141·6-s − 1.73·7-s + 0.476·8-s + 0.333·9-s − 0.109·10-s + 1.55·11-s + 0.542·12-s − 1.30·13-s + 0.425·14-s − 0.258·15-s + 0.822·16-s + 0.908·17-s − 0.0819·18-s − 0.374·19-s − 0.420·20-s + 0.999·21-s − 0.383·22-s − 0.844·23-s − 0.275·24-s + 0.200·25-s + 0.320·26-s − 0.192·27-s + 1.62·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6015\)    =    \(3 \cdot 5 \cdot 401\)
Sign: $1$
Analytic conductor: \(48.0300\)
Root analytic conductor: \(6.93036\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6015,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6011408388\)
\(L(\frac12)\) \(\approx\) \(0.6011408388\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 - T \)
401 \( 1 + T \)
good2 \( 1 + 0.347T + 2T^{2} \)
7 \( 1 + 4.57T + 7T^{2} \)
11 \( 1 - 5.17T + 11T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 - 3.74T + 17T^{2} \)
19 \( 1 + 1.63T + 19T^{2} \)
23 \( 1 + 4.05T + 23T^{2} \)
29 \( 1 + 4.36T + 29T^{2} \)
31 \( 1 - 0.267T + 31T^{2} \)
37 \( 1 + 2.19T + 37T^{2} \)
41 \( 1 - 2.00T + 41T^{2} \)
43 \( 1 - 3.41T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 7.89T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 2.07T + 61T^{2} \)
67 \( 1 - 7.80T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 + 3.67T + 73T^{2} \)
79 \( 1 + 17.1T + 79T^{2} \)
83 \( 1 + 2.74T + 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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

−8.158511133151189704542914508789, −7.11400214828245750425463876237, −6.74995595168552703450766935263, −5.82974491621603311232218100174, −5.45466347515848706828384759033, −4.28678016708427896211620189281, −3.83370764842131983095271089438, −2.89177291067313513589937086699, −1.59448685542242147797035589710, −0.44794652462043877397975332926, 0.44794652462043877397975332926, 1.59448685542242147797035589710, 2.89177291067313513589937086699, 3.83370764842131983095271089438, 4.28678016708427896211620189281, 5.45466347515848706828384759033, 5.82974491621603311232218100174, 6.74995595168552703450766935263, 7.11400214828245750425463876237, 8.158511133151189704542914508789

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