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Properties

Label 2-3024-28.27-c1-0-40
Degree $2$
Conductor $3024$
Sign $0.188 + 0.981i$
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  + (−2.5 − 0.866i)7-s + 1.73i·13-s + 19-s + 5·25-s − 4·31-s + 37-s − 10.3i·43-s + (5.5 + 4.33i)49-s − 8.66i·61-s − 12.1i·67-s − 1.73i·73-s − 12.1i·79-s + (1.49 − 4.33i)91-s − 19.0i·97-s − 7·103-s + ⋯
L(s)  = 1  + (−0.944 − 0.327i)7-s + 0.480i·13-s + 0.229·19-s + 25-s − 0.718·31-s + 0.164·37-s − 1.58i·43-s + (0.785 + 0.618i)49-s − 1.10i·61-s − 1.48i·67-s − 0.202i·73-s − 1.36i·79-s + (0.157 − 0.453i)91-s − 1.93i·97-s − 0.689·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.188 + 0.981i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.188 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.185078736\)
\(L(\frac12)\) \(\approx\) \(1.185078736\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 19.0iT - 97T^{2} \)
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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

−8.730946668540202363779368830964, −7.68864693948181262542318317339, −7.00087654990579005132539029906, −6.40583952799669887976218667675, −5.54080668261508826367836218049, −4.63284382844607820728718813521, −3.71530613657385921574732772242, −2.99707027982598922428588140467, −1.83324001589955420355968465519, −0.42674313281574924849884234814, 1.04762214568876821653473194168, 2.50104724660868707014546709584, 3.19200314612979602348259899630, 4.11460996600220646220158461440, 5.15488220060993446885474017736, 5.85617729892471095589469647032, 6.63235754814692409934266158721, 7.31236684886691527185828898520, 8.196928458065173802739480179206, 8.940862497842335180257010594715

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