Properties

Label 2-8015-1.1-c1-0-404
Degree $2$
Conductor $8015$
Sign $-1$
Analytic cond. $64.0000$
Root an. cond. $8.00000$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.87·2-s − 0.460·3-s + 1.51·4-s + 5-s − 0.863·6-s − 7-s − 0.901·8-s − 2.78·9-s + 1.87·10-s + 0.923·11-s − 0.699·12-s + 3.86·13-s − 1.87·14-s − 0.460·15-s − 4.73·16-s − 0.595·17-s − 5.23·18-s + 1.84·19-s + 1.51·20-s + 0.460·21-s + 1.73·22-s − 2.62·23-s + 0.415·24-s + 25-s + 7.24·26-s + 2.66·27-s − 1.51·28-s + ⋯
L(s)  = 1  + 1.32·2-s − 0.265·3-s + 0.759·4-s + 0.447·5-s − 0.352·6-s − 0.377·7-s − 0.318·8-s − 0.929·9-s + 0.593·10-s + 0.278·11-s − 0.201·12-s + 1.07·13-s − 0.501·14-s − 0.118·15-s − 1.18·16-s − 0.144·17-s − 1.23·18-s + 0.422·19-s + 0.339·20-s + 0.100·21-s + 0.369·22-s − 0.546·23-s + 0.0847·24-s + 0.200·25-s + 1.42·26-s + 0.512·27-s − 0.287·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8015\)    =    \(5 \cdot 7 \cdot 229\)
Sign: $-1$
Analytic conductor: \(64.0000\)
Root analytic conductor: \(8.00000\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8015,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 + T \)
229 \( 1 - T \)
good2 \( 1 - 1.87T + 2T^{2} \)
3 \( 1 + 0.460T + 3T^{2} \)
11 \( 1 - 0.923T + 11T^{2} \)
13 \( 1 - 3.86T + 13T^{2} \)
17 \( 1 + 0.595T + 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 + 8.17T + 29T^{2} \)
31 \( 1 - 5.02T + 31T^{2} \)
37 \( 1 - 6.81T + 37T^{2} \)
41 \( 1 - 4.87T + 41T^{2} \)
43 \( 1 + 7.51T + 43T^{2} \)
47 \( 1 + 4.68T + 47T^{2} \)
53 \( 1 + 7.86T + 53T^{2} \)
59 \( 1 - 7.18T + 59T^{2} \)
61 \( 1 + 4.02T + 61T^{2} \)
67 \( 1 + 2.34T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + 5.21T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 9.04T + 83T^{2} \)
89 \( 1 + 8.68T + 89T^{2} \)
97 \( 1 - 2.48T + 97T^{2} \)
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

−7.19386271106773527452277307223, −6.32821877113810805232616678496, −5.98944268402674460567152319872, −5.53436875894826876245779196565, −4.66449660966715738832215220324, −3.95705781589114460330398906652, −3.21242537616123579146617319950, −2.62268075576523605630464290706, −1.46033610897761698000680657237, 0, 1.46033610897761698000680657237, 2.62268075576523605630464290706, 3.21242537616123579146617319950, 3.95705781589114460330398906652, 4.66449660966715738832215220324, 5.53436875894826876245779196565, 5.98944268402674460567152319872, 6.32821877113810805232616678496, 7.19386271106773527452277307223

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