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Properties

Label 2-3024-63.47-c1-0-29
Degree $2$
Conductor $3024$
Sign $0.483 + 0.875i$
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  + 0.203·5-s + (1.27 − 2.32i)7-s − 4.46i·11-s + (1.25 − 0.725i)13-s + (1.60 + 2.78i)17-s + (6.20 + 3.58i)19-s + 1.26i·23-s − 4.95·25-s + (0.944 + 0.545i)29-s + (5.60 + 3.23i)31-s + (0.258 − 0.471i)35-s + (3.02 − 5.24i)37-s + (0.370 + 0.642i)41-s + (4.69 − 8.13i)43-s + (−0.0465 − 0.0806i)47-s + ⋯
L(s)  = 1  + 0.0908·5-s + (0.480 − 0.876i)7-s − 1.34i·11-s + (0.348 − 0.201i)13-s + (0.389 + 0.674i)17-s + (1.42 + 0.821i)19-s + 0.264i·23-s − 0.991·25-s + (0.175 + 0.101i)29-s + (1.00 + 0.580i)31-s + (0.0436 − 0.0796i)35-s + (0.497 − 0.862i)37-s + (0.0578 + 0.100i)41-s + (0.716 − 1.24i)43-s + (−0.00679 − 0.0117i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.483 + 0.875i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.068025106\)
\(L(\frac12)\) \(\approx\) \(2.068025106\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 + 2.32i)T \)
good5 \( 1 - 0.203T + 5T^{2} \)
11 \( 1 + 4.46iT - 11T^{2} \)
13 \( 1 + (-1.25 + 0.725i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.60 - 2.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.20 - 3.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.26iT - 23T^{2} \)
29 \( 1 + (-0.944 - 0.545i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.60 - 3.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.02 + 5.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.370 - 0.642i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.69 + 8.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0465 + 0.0806i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.35 - 5.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.16 + 8.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.34 - 4.24i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.02 - 6.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (-0.984 + 0.568i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.86 + 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.29 - 3.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.52 + 6.10i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.17 + 1.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.450004777886337010374265238625, −7.84815187246474011539161656162, −7.30398810469382808870919576696, −6.06962710262778756015756067430, −5.75202026024802071038175693156, −4.68597301336130291931518369067, −3.68862157026992443247312736432, −3.19293169892384376605374674271, −1.66342899939003185132618141687, −0.75050993152363673763633286404, 1.19597832548486293511197087418, 2.30006303095905529861386923263, 3.04248922367032301174699156753, 4.39525698716144850720136772341, 4.91782951999300474313914949242, 5.76382040026805545436076527322, 6.57470486016206133487599120716, 7.49278189349375767664417885595, 7.976863912437286589781556177837, 8.940121245290112062990449401480

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