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Properties

Label 2-3024-63.16-c1-0-4
Degree $2$
Conductor $3024$
Sign $0.445 - 0.895i$
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  − 3.19·5-s + (−2.61 − 0.415i)7-s − 2.28·11-s + (−0.675 − 1.16i)13-s + (−2.21 − 3.83i)17-s + (3.69 − 6.39i)19-s − 6.46·23-s + 5.20·25-s + (1.06 − 1.83i)29-s + (−0.316 + 0.547i)31-s + (8.34 + 1.32i)35-s + (1.92 − 3.34i)37-s + (5.05 + 8.74i)41-s + (−4.24 + 7.35i)43-s + (−3.26 − 5.65i)47-s + ⋯
L(s)  = 1  − 1.42·5-s + (−0.987 − 0.157i)7-s − 0.688·11-s + (−0.187 − 0.324i)13-s + (−0.537 − 0.930i)17-s + (0.847 − 1.46i)19-s − 1.34·23-s + 1.04·25-s + (0.197 − 0.341i)29-s + (−0.0567 + 0.0983i)31-s + (1.41 + 0.224i)35-s + (0.317 − 0.549i)37-s + (0.788 + 1.36i)41-s + (−0.647 + 1.12i)43-s + (−0.476 − 0.825i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.445 - 0.895i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.445 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3774839858\)
\(L(\frac12)\) \(\approx\) \(0.3774839858\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.61 + 0.415i)T \)
good5 \( 1 + 3.19T + 5T^{2} \)
11 \( 1 + 2.28T + 11T^{2} \)
13 \( 1 + (0.675 + 1.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.21 + 3.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.69 + 6.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.46T + 23T^{2} \)
29 \( 1 + (-1.06 + 1.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.316 - 0.547i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.92 + 3.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.05 - 8.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.24 - 7.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.26 + 5.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.39 + 4.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.10 - 5.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.45 - 7.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.50 - 2.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + (-4.36 - 7.56i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.938 + 1.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.00 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.65 - 4.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.44 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.790052888162555392839515190923, −7.964591562248771370045937496338, −7.39611677579626929504442931611, −6.81448379893324503926699360782, −5.83831996034507335121296700539, −4.80677522629729065788107229942, −4.18281918676133622048713045187, −3.16539353178262435038432026322, −2.62948057290751547262256720886, −0.63832101184923093169075548022, 0.19902864834036727776704805884, 1.91321779900786511706623180075, 3.18835783041500913470738615704, 3.76397855920237828047016263820, 4.48649371249446947320922068977, 5.64692233360532808479171140074, 6.30727639021474981064814151547, 7.25140797065882206015707109048, 7.86791131275700784185516439341, 8.398127662489960068287762027567

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