Properties

Label 2-8024-1.1-c1-0-50
Degree $2$
Conductor $8024$
Sign $1$
Analytic cond. $64.0719$
Root an. cond. $8.00449$
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  + 1.84·3-s − 2.21·5-s − 0.448·7-s + 0.393·9-s − 0.342·11-s − 5.00·13-s − 4.08·15-s − 17-s + 3.51·19-s − 0.825·21-s + 0.475·23-s − 0.0819·25-s − 4.80·27-s − 4.16·29-s + 10.2·31-s − 0.630·33-s + 0.993·35-s + 10.6·37-s − 9.21·39-s + 6.40·41-s − 10.3·43-s − 0.872·45-s − 6.91·47-s − 6.79·49-s − 1.84·51-s + 7.34·53-s + 0.758·55-s + ⋯
L(s)  = 1  + 1.06·3-s − 0.991·5-s − 0.169·7-s + 0.131·9-s − 0.103·11-s − 1.38·13-s − 1.05·15-s − 0.242·17-s + 0.807·19-s − 0.180·21-s + 0.0991·23-s − 0.0163·25-s − 0.924·27-s − 0.772·29-s + 1.84·31-s − 0.109·33-s + 0.168·35-s + 1.75·37-s − 1.47·39-s + 1.00·41-s − 1.58·43-s − 0.130·45-s − 1.00·47-s − 0.971·49-s − 0.257·51-s + 1.00·53-s + 0.102·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8024\)    =    \(2^{3} \cdot 17 \cdot 59\)
Sign: $1$
Analytic conductor: \(64.0719\)
Root analytic conductor: \(8.00449\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.713775009\)
\(L(\frac12)\) \(\approx\) \(1.713775009\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 - T \)
good3 \( 1 - 1.84T + 3T^{2} \)
5 \( 1 + 2.21T + 5T^{2} \)
7 \( 1 + 0.448T + 7T^{2} \)
11 \( 1 + 0.342T + 11T^{2} \)
13 \( 1 + 5.00T + 13T^{2} \)
19 \( 1 - 3.51T + 19T^{2} \)
23 \( 1 - 0.475T + 23T^{2} \)
29 \( 1 + 4.16T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 - 6.40T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 6.91T + 47T^{2} \)
53 \( 1 - 7.34T + 53T^{2} \)
61 \( 1 - 9.00T + 61T^{2} \)
67 \( 1 - 4.56T + 67T^{2} \)
71 \( 1 - 4.13T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 - 0.0598T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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

−7.926656741859796819415161012562, −7.39190553679815128318916873181, −6.68340884141635182420630727707, −5.71619490658482354362606020838, −4.81450286205952313367340040554, −4.21684408567396792269366576215, −3.34583142241460308568373668847, −2.81527345621802815634416262975, −2.04411264946115693296788911959, −0.58565341444679712214008075895, 0.58565341444679712214008075895, 2.04411264946115693296788911959, 2.81527345621802815634416262975, 3.34583142241460308568373668847, 4.21684408567396792269366576215, 4.81450286205952313367340040554, 5.71619490658482354362606020838, 6.68340884141635182420630727707, 7.39190553679815128318916873181, 7.926656741859796819415161012562

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