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Properties

Label 2-3024-63.47-c1-0-13
Degree $2$
Conductor $3024$
Sign $0.569 - 0.822i$
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  + 1.81·5-s + (−1.69 − 2.03i)7-s − 0.255i·11-s + (−5.77 + 3.33i)13-s + (1.99 + 3.46i)17-s + (−1.24 − 0.719i)19-s − 5.66i·23-s − 1.71·25-s + (4.18 + 2.41i)29-s + (8.80 + 5.08i)31-s + (−3.06 − 3.68i)35-s + (−1.65 + 2.86i)37-s + (5.10 + 8.83i)41-s + (−1.12 + 1.94i)43-s + (5.97 + 10.3i)47-s + ⋯
L(s)  = 1  + 0.810·5-s + (−0.639 − 0.768i)7-s − 0.0771i·11-s + (−1.60 + 0.925i)13-s + (0.484 + 0.839i)17-s + (−0.286 − 0.165i)19-s − 1.18i·23-s − 0.343·25-s + (0.777 + 0.448i)29-s + (1.58 + 0.913i)31-s + (−0.518 − 0.622i)35-s + (−0.272 + 0.471i)37-s + (0.796 + 1.37i)41-s + (−0.171 + 0.296i)43-s + (0.870 + 1.50i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.569 - 0.822i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.569 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.547557848\)
\(L(\frac12)\) \(\approx\) \(1.547557848\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 2.03i)T \)
good5 \( 1 - 1.81T + 5T^{2} \)
11 \( 1 + 0.255iT - 11T^{2} \)
13 \( 1 + (5.77 - 3.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.99 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.24 + 0.719i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.66iT - 23T^{2} \)
29 \( 1 + (-4.18 - 2.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.80 - 5.08i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.65 - 2.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.10 - 8.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.12 - 1.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.97 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.97 + 2.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.55 - 4.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.60 + 4.96i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.962 + 1.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.31iT - 71T^{2} \)
73 \( 1 + (2.47 - 1.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.83 + 3.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.68 + 4.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.378 + 0.655i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.21 - 2.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.910749293431126479647541878306, −8.095217476658020993853834533088, −7.19396900875344445340484928961, −6.53102891296475539412643391381, −6.02610340031068606940525935589, −4.77438713283120703532537848571, −4.35708725111576300812821696857, −3.07126273747012988148015656856, −2.30533422174531729545801812972, −1.08170676587439134707606168975, 0.52748463593140872382943053407, 2.24975658784872541279719310971, 2.63652004809259104986019442566, 3.76201927140073428771763192599, 5.02957008219580851415310482542, 5.53086558212277843955028816876, 6.17900175107971886663618694436, 7.17268967949803760812039948762, 7.75578622964996517030727084186, 8.736927108290848855157262950356

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