↑

Properties

Label 2-3024-63.41-c1-0-36
Degree $2$
Conductor $3024$
Sign $0.340 + 0.940i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.79 − 3.10i)5-s + (2.32 − 1.26i)7-s + (−4.38 + 2.53i)11-s + (5.48 + 3.16i)13-s + 0.256·17-s − 6.71i·19-s + (0.138 + 0.0802i)23-s + (−3.92 − 6.80i)25-s + (1.17 − 0.675i)29-s + (3.01 + 1.74i)31-s + (0.251 − 9.48i)35-s + 9.37·37-s + (2.31 − 4.01i)41-s + (2.70 + 4.68i)43-s + (1.50 + 2.60i)47-s + ⋯
L(s)  = 1  + (0.801 − 1.38i)5-s + (0.878 − 0.476i)7-s + (−1.32 + 0.763i)11-s + (1.52 + 0.879i)13-s + 0.0622·17-s − 1.54i·19-s + (0.0289 + 0.0167i)23-s + (−0.785 − 1.36i)25-s + (0.217 − 0.125i)29-s + (0.541 + 0.312i)31-s + (0.0424 − 1.60i)35-s + 1.54·37-s + (0.362 − 0.627i)41-s + (0.412 + 0.714i)43-s + (0.219 + 0.380i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.340 + 0.940i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.340 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.438798487\)
\(L(\frac12)\) \(\approx\) \(2.438798487\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.32 + 1.26i)T \)
good5 \( 1 + (-1.79 + 3.10i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.38 - 2.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.48 - 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.256T + 17T^{2} \)
19 \( 1 + 6.71iT - 19T^{2} \)
23 \( 1 + (-0.138 - 0.0802i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.17 + 0.675i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.01 - 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.37T + 37T^{2} \)
41 \( 1 + (-2.31 + 4.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.70 - 4.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.50 - 2.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.35iT - 53T^{2} \)
59 \( 1 + (-2.32 + 4.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.50 - 1.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.52 - 7.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.74iT - 71T^{2} \)
73 \( 1 + 7.77iT - 73T^{2} \)
79 \( 1 + (6.10 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.90 + 3.30i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.79T + 89T^{2} \)
97 \( 1 + (11.0 - 6.35i)T + (48.5 - 84.0i)T^{2} \)
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

−8.611244068583996677000454563331, −7.995147638620889750772361598431, −7.15405070733353075511685065144, −6.16930758077614572676569425159, −5.37471319138698521382559732650, −4.63715402819081981761252005391, −4.27757921521080912223566077601, −2.65098679242176406939786817778, −1.69717929330951279394435290783, −0.850641419901943897078680541720, 1.27569535025282361454735846293, 2.47705257674509615513199199658, 3.01834027797100593793042966185, 4.01466334917154218936552577327, 5.41007427295360463283322641410, 5.84458386787364407077491883578, 6.31874098634523466173353243508, 7.57614165852196994002772628067, 8.062102232038731562185212083054, 8.663853887177119651297169430577

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