↑

Properties

Label 2-84e2-1.1-c1-0-74
Degree $2$
Conductor $7056$
Sign $-1$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·13-s + 4·17-s − 4·19-s + 4·23-s − 5·25-s − 2·29-s + 8·31-s − 6·37-s + 12·41-s − 4·43-s + 8·47-s − 6·53-s − 12·59-s − 4·61-s + 4·67-s − 12·71-s − 8·73-s + 16·79-s + 4·83-s + 4·89-s − 16·97-s − 8·101-s − 8·103-s + 8·107-s − 14·109-s − 2·113-s + ⋯
L(s)  = 1  − 1.10·13-s + 0.970·17-s − 0.917·19-s + 0.834·23-s − 25-s − 0.371·29-s + 1.43·31-s − 0.986·37-s + 1.87·41-s − 0.609·43-s + 1.16·47-s − 0.824·53-s − 1.56·59-s − 0.512·61-s + 0.488·67-s − 1.42·71-s − 0.936·73-s + 1.80·79-s + 0.439·83-s + 0.423·89-s − 1.62·97-s − 0.796·101-s − 0.788·103-s + 0.773·107-s − 1.34·109-s − 0.188·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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

−7.70555464733637830966904621801, −6.93075127573152302851245219076, −6.18508362209800088128659462191, −5.47899885347988873314570699626, −4.72051468542461878090774249710, −4.05040289726970139512021818636, −3.05497446481360500202092438602, −2.35921267304058176004459520482, −1.28728303516551820824700676795, 0, 1.28728303516551820824700676795, 2.35921267304058176004459520482, 3.05497446481360500202092438602, 4.05040289726970139512021818636, 4.72051468542461878090774249710, 5.47899885347988873314570699626, 6.18508362209800088128659462191, 6.93075127573152302851245219076, 7.70555464733637830966904621801

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