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Properties

Label 2-3024-63.20-c1-0-28
Degree $2$
Conductor $3024$
Sign $-0.0303 + 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.26 + 2.19i)5-s + (−1.98 + 1.74i)7-s + (−5.68 − 3.28i)11-s + (−5.13 + 2.96i)13-s + 3.52·17-s − 0.261i·19-s + (2.89 − 1.67i)23-s + (−0.718 + 1.24i)25-s + (1.00 + 0.582i)29-s + (1.69 − 0.977i)31-s + (−6.36 − 2.14i)35-s − 1.16·37-s + (−2.85 − 4.94i)41-s + (−2.67 + 4.63i)43-s + (3.79 − 6.57i)47-s + ⋯
L(s)  = 1  + (0.567 + 0.982i)5-s + (−0.750 + 0.660i)7-s + (−1.71 − 0.989i)11-s + (−1.42 + 0.822i)13-s + 0.855·17-s − 0.0599i·19-s + (0.604 − 0.349i)23-s + (−0.143 + 0.248i)25-s + (0.187 + 0.108i)29-s + (0.304 − 0.175i)31-s + (−1.07 − 0.362i)35-s − 0.191·37-s + (−0.446 − 0.773i)41-s + (−0.407 + 0.706i)43-s + (0.553 − 0.959i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0303 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0303 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.0303 + 0.999i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.0303 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5256182972\)
\(L(\frac12)\) \(\approx\) \(0.5256182972\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.98 - 1.74i)T \)
good5 \( 1 + (-1.26 - 2.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.68 + 3.28i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.13 - 2.96i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 + 0.261iT - 19T^{2} \)
23 \( 1 + (-2.89 + 1.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.00 - 0.582i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.69 + 0.977i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.16T + 37T^{2} \)
41 \( 1 + (2.85 + 4.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.67 - 4.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.79 + 6.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.54iT - 53T^{2} \)
59 \( 1 + (5.47 + 9.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.53 - 3.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.54 + 7.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + 2.72iT - 73T^{2} \)
79 \( 1 + (-0.652 + 1.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.53 + 7.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (9.95 + 5.74i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.548875216677354236327242032021, −7.70927341074002119965736509581, −6.94718560065827491507758619449, −6.29814231031222729670055495831, −5.47776472685774807289264901401, −4.92329304102905426545270682697, −3.40351822420601612691143465796, −2.75814051583335132412673878165, −2.20915163025225859849140538458, −0.17130933159710709336321805061, 1.08773397207064079276764397034, 2.42913506920190213541089787682, 3.14183452754613344521148075161, 4.47988428995947024712582013719, 5.15296713430213295879087301968, 5.55838325280929864770269457948, 6.76649395786956916269507281632, 7.63289812820673555753348531404, 7.87475322190844402576662640222, 9.106089632054895974291495750066

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