Properties

Label 2-588-1.1-c3-0-1
Degree $2$
Conductor $588$
Sign $1$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 14·5-s + 9·9-s + 4·11-s − 54·13-s + 42·15-s + 14·17-s − 92·19-s − 152·23-s + 71·25-s − 27·27-s − 106·29-s + 144·31-s − 12·33-s + 158·37-s + 162·39-s + 390·41-s − 508·43-s − 126·45-s + 528·47-s − 42·51-s + 606·53-s − 56·55-s + 276·57-s + 364·59-s − 678·61-s + 756·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.25·5-s + 1/3·9-s + 0.109·11-s − 1.15·13-s + 0.722·15-s + 0.199·17-s − 1.11·19-s − 1.37·23-s + 0.567·25-s − 0.192·27-s − 0.678·29-s + 0.834·31-s − 0.0633·33-s + 0.702·37-s + 0.665·39-s + 1.48·41-s − 1.80·43-s − 0.417·45-s + 1.63·47-s − 0.115·51-s + 1.57·53-s − 0.137·55-s + 0.641·57-s + 0.803·59-s − 1.42·61-s + 1.44·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{588} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6990570068\)
\(L(\frac12)\) \(\approx\) \(0.6990570068\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
7 \( 1 \)
good5 \( 1 + 14 T + p^{3} T^{2} \)
11 \( 1 - 4 T + p^{3} T^{2} \)
13 \( 1 + 54 T + p^{3} T^{2} \)
17 \( 1 - 14 T + p^{3} T^{2} \)
19 \( 1 + 92 T + p^{3} T^{2} \)
23 \( 1 + 152 T + p^{3} T^{2} \)
29 \( 1 + 106 T + p^{3} T^{2} \)
31 \( 1 - 144 T + p^{3} T^{2} \)
37 \( 1 - 158 T + p^{3} T^{2} \)
41 \( 1 - 390 T + p^{3} T^{2} \)
43 \( 1 + 508 T + p^{3} T^{2} \)
47 \( 1 - 528 T + p^{3} T^{2} \)
53 \( 1 - 606 T + p^{3} T^{2} \)
59 \( 1 - 364 T + p^{3} T^{2} \)
61 \( 1 + 678 T + p^{3} T^{2} \)
67 \( 1 - 844 T + p^{3} T^{2} \)
71 \( 1 + 8 T + p^{3} T^{2} \)
73 \( 1 - 422 T + p^{3} T^{2} \)
79 \( 1 - 384 T + p^{3} T^{2} \)
83 \( 1 - 548 T + p^{3} T^{2} \)
89 \( 1 + 1194 T + p^{3} T^{2} \)
97 \( 1 - 1502 T + p^{3} T^{2} \)
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

−10.39697519221273219773165377423, −9.540681941358917864728672066726, −8.307119842683330085741499981555, −7.64623699531748283936543360696, −6.77040263465354876446366978891, −5.69685198427605566396711113184, −4.51524023592095186587561893035, −3.86094642201379620048109741390, −2.31271776923834928150750858762, −0.49455119663587675923333046082, 0.49455119663587675923333046082, 2.31271776923834928150750858762, 3.86094642201379620048109741390, 4.51524023592095186587561893035, 5.69685198427605566396711113184, 6.77040263465354876446366978891, 7.64623699531748283936543360696, 8.307119842683330085741499981555, 9.540681941358917864728672066726, 10.39697519221273219773165377423

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