Properties

Label 2-8048-1.1-c1-0-242
Degree $2$
Conductor $8048$
Sign $-1$
Analytic cond. $64.2636$
Root an. cond. $8.01645$
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  + 2.22·3-s − 1.09·5-s + 1.95·7-s + 1.94·9-s + 2.63·11-s − 0.307·13-s − 2.43·15-s − 5.78·17-s − 0.355·19-s + 4.34·21-s + 1.82·23-s − 3.79·25-s − 2.35·27-s − 9.46·29-s − 8.77·31-s + 5.84·33-s − 2.14·35-s + 0.369·37-s − 0.684·39-s − 5.32·41-s − 4.57·43-s − 2.12·45-s + 4.16·47-s − 3.17·49-s − 12.8·51-s + 3.30·53-s − 2.88·55-s + ⋯
L(s)  = 1  + 1.28·3-s − 0.490·5-s + 0.739·7-s + 0.647·9-s + 0.793·11-s − 0.0853·13-s − 0.629·15-s − 1.40·17-s − 0.0815·19-s + 0.948·21-s + 0.379·23-s − 0.759·25-s − 0.452·27-s − 1.75·29-s − 1.57·31-s + 1.01·33-s − 0.362·35-s + 0.0608·37-s − 0.109·39-s − 0.832·41-s − 0.697·43-s − 0.317·45-s + 0.606·47-s − 0.453·49-s − 1.79·51-s + 0.454·53-s − 0.388·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8048\)    =    \(2^{4} \cdot 503\)
Sign: $-1$
Analytic conductor: \(64.2636\)
Root analytic conductor: \(8.01645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8048,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
503 \( 1 - T \)
good3 \( 1 - 2.22T + 3T^{2} \)
5 \( 1 + 1.09T + 5T^{2} \)
7 \( 1 - 1.95T + 7T^{2} \)
11 \( 1 - 2.63T + 11T^{2} \)
13 \( 1 + 0.307T + 13T^{2} \)
17 \( 1 + 5.78T + 17T^{2} \)
19 \( 1 + 0.355T + 19T^{2} \)
23 \( 1 - 1.82T + 23T^{2} \)
29 \( 1 + 9.46T + 29T^{2} \)
31 \( 1 + 8.77T + 31T^{2} \)
37 \( 1 - 0.369T + 37T^{2} \)
41 \( 1 + 5.32T + 41T^{2} \)
43 \( 1 + 4.57T + 43T^{2} \)
47 \( 1 - 4.16T + 47T^{2} \)
53 \( 1 - 3.30T + 53T^{2} \)
59 \( 1 - 7.66T + 59T^{2} \)
61 \( 1 + 8.11T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 9.39T + 71T^{2} \)
73 \( 1 - 6.49T + 73T^{2} \)
79 \( 1 - 5.39T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 3.57T + 89T^{2} \)
97 \( 1 - 6.26T + 97T^{2} \)
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

−7.45411965874059342820217471374, −7.19471637472731921833749535706, −6.16494022870483255105804649002, −5.30233103611344369007225191085, −4.37759274750289868281445447352, −3.84779639684692345108125160087, −3.20748847467656406795661476718, −2.08458979182525022920408211935, −1.69724682688250925108180004380, 0, 1.69724682688250925108180004380, 2.08458979182525022920408211935, 3.20748847467656406795661476718, 3.84779639684692345108125160087, 4.37759274750289868281445447352, 5.30233103611344369007225191085, 6.16494022870483255105804649002, 7.19471637472731921833749535706, 7.45411965874059342820217471374

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