↑

Properties

Label 2-3024-21.2-c0-0-3
Degree $2$
Conductor $3024$
Sign $0.0633 + 0.997i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 13-s + (1 − 1.73i)19-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)37-s + 43-s + 49-s + (0.5 − 0.866i)61-s + (−0.5 − 0.866i)67-s + (−1 − 1.73i)73-s + (−0.5 + 0.866i)79-s + 91-s − 97-s + (−0.5 + 0.866i)103-s + ⋯
L(s)  = 1  − 7-s − 13-s + (1 − 1.73i)19-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)37-s + 43-s + 49-s + (0.5 − 0.866i)61-s + (−0.5 − 0.866i)67-s + (−1 − 1.73i)73-s + (−0.5 + 0.866i)79-s + 91-s − 97-s + (−0.5 + 0.866i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8113405034\)
\(L(\frac12)\) \(\approx\) \(0.8113405034\)
\(L(1)\) 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 \( 1 + T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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

−8.992232123888958455299214067070, −7.77345334186964170867152331616, −7.27291396780939132508478232796, −6.50345586149872442531138677076, −5.71191455199836150335955269496, −4.85723124621045564033584194109, −3.99894160901838265591739202127, −2.95695629317373301599398233559, −2.29472795235961351843387735590, −0.50439234841334273830188845677, 1.41811173126597591421305246405, 2.70927041221995102863183592306, 3.46303940723958090843753790714, 4.31711393591598757349870730914, 5.48394997037234701269515888090, 5.88786788612403676145807705832, 7.00812752671467529549034396971, 7.43002853787701888094006242298, 8.321924845540835422728063984327, 9.216621866562031696357892141403

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