Properties

Label 2-3024-63.59-c1-0-17
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: $\chi_{3024} (17, \cdot )$
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

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

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