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Properties

Label 2-3024-63.47-c1-0-22
Degree $2$
Conductor $3024$
Sign $0.992 + 0.121i$
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  − 2.86·5-s + (1.83 + 1.90i)7-s − 2.71i·11-s + (3.18 − 1.84i)13-s + (3.22 + 5.58i)17-s + (−2.73 − 1.58i)19-s − 2.99i·23-s + 3.22·25-s + (2.48 + 1.43i)29-s + (−8.26 − 4.77i)31-s + (−5.25 − 5.47i)35-s + (−1.70 + 2.95i)37-s + (−0.794 − 1.37i)41-s + (4.67 − 8.10i)43-s + (5.65 + 9.79i)47-s + ⋯
L(s)  = 1  − 1.28·5-s + (0.692 + 0.721i)7-s − 0.817i·11-s + (0.884 − 0.510i)13-s + (0.781 + 1.35i)17-s + (−0.628 − 0.362i)19-s − 0.623i·23-s + 0.645·25-s + (0.461 + 0.266i)29-s + (−1.48 − 0.857i)31-s + (−0.888 − 0.925i)35-s + (−0.280 + 0.485i)37-s + (−0.124 − 0.214i)41-s + (0.713 − 1.23i)43-s + (0.824 + 1.42i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.992 + 0.121i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.992 + 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.496924648\)
\(L(\frac12)\) \(\approx\) \(1.496924648\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.83 - 1.90i)T \)
good5 \( 1 + 2.86T + 5T^{2} \)
11 \( 1 + 2.71iT - 11T^{2} \)
13 \( 1 + (-3.18 + 1.84i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.22 - 5.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.73 + 1.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.99iT - 23T^{2} \)
29 \( 1 + (-2.48 - 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.26 + 4.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.70 - 2.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.794 + 1.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.67 + 8.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.65 - 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.16 + 1.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.33 - 7.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.566 - 0.327i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.86 + 6.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 + (-11.0 + 6.39i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.59 - 4.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.92 + 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.14 + 5.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.2 - 7.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.655641209963304753009419547332, −7.967533777005830517125304013488, −7.54468964506979306305723646954, −6.23972165599671801247808402080, −5.78213952923042646459257379707, −4.76273258547834700987573971658, −3.85488565423702489343377754333, −3.30656079671357081472472745456, −2.02834759629311401063495587823, −0.70779740913444206277929407212, 0.810653326065845607423329869835, 1.96487766515983164964556445923, 3.41236512268599074618834127725, 3.98916063080114809823079039067, 4.70891266812666510223472411648, 5.52398784666093801595315778415, 6.81294903518175835479293953645, 7.30755083118102382613358734618, 7.88470739470540466642343877326, 8.576664088805321251108691579369

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