Properties

Label 2-8024-1.1-c1-0-174
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.45·3-s + 2.04·5-s + 2.58·7-s + 8.93·9-s − 2.30·11-s − 4.87·13-s + 7.05·15-s + 17-s + 4.97·19-s + 8.93·21-s + 3.63·23-s − 0.832·25-s + 20.4·27-s + 8.15·29-s − 7.13·31-s − 7.95·33-s + 5.27·35-s + 5.94·37-s − 16.8·39-s + 0.901·41-s − 3.72·43-s + 18.2·45-s + 2.64·47-s − 0.312·49-s + 3.45·51-s − 13.9·53-s − 4.70·55-s + ⋯
L(s)  = 1  + 1.99·3-s + 0.912·5-s + 0.977·7-s + 2.97·9-s − 0.694·11-s − 1.35·13-s + 1.82·15-s + 0.242·17-s + 1.14·19-s + 1.94·21-s + 0.758·23-s − 0.166·25-s + 3.94·27-s + 1.51·29-s − 1.28·31-s − 1.38·33-s + 0.892·35-s + 0.976·37-s − 2.69·39-s + 0.140·41-s − 0.568·43-s + 2.71·45-s + 0.385·47-s − 0.0445·49-s + 0.483·51-s − 1.91·53-s − 0.634·55-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

\(L(1)\) \(\approx\) \(6.305243680\)
\(L(\frac12)\) \(\approx\) \(6.305243680\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.45T + 3T^{2} \)
5 \( 1 - 2.04T + 5T^{2} \)
7 \( 1 - 2.58T + 7T^{2} \)
11 \( 1 + 2.30T + 11T^{2} \)
13 \( 1 + 4.87T + 13T^{2} \)
19 \( 1 - 4.97T + 19T^{2} \)
23 \( 1 - 3.63T + 23T^{2} \)
29 \( 1 - 8.15T + 29T^{2} \)
31 \( 1 + 7.13T + 31T^{2} \)
37 \( 1 - 5.94T + 37T^{2} \)
41 \( 1 - 0.901T + 41T^{2} \)
43 \( 1 + 3.72T + 43T^{2} \)
47 \( 1 - 2.64T + 47T^{2} \)
53 \( 1 + 13.9T + 53T^{2} \)
61 \( 1 - 8.07T + 61T^{2} \)
67 \( 1 + 4.93T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + 2.52T + 89T^{2} \)
97 \( 1 + 3.18T + 97T^{2} \)
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

−7.944084479283443707378154103302, −7.36991441840429499620087924086, −6.82734277449067989185411343900, −5.53908075305154544892275863319, −4.88961731027333857242399738768, −4.30998367837955922586605630439, −3.11531380212776818920212112806, −2.72034132395918145839973628117, −1.95093788878367569971755390962, −1.27359428686018079185178976303, 1.27359428686018079185178976303, 1.95093788878367569971755390962, 2.72034132395918145839973628117, 3.11531380212776818920212112806, 4.30998367837955922586605630439, 4.88961731027333857242399738768, 5.53908075305154544892275863319, 6.82734277449067989185411343900, 7.36991441840429499620087924086, 7.944084479283443707378154103302

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