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Properties

Label 2-3024-63.41-c1-0-43
Degree $2$
Conductor $3024$
Sign $-0.739 + 0.673i$
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  + (0.965 − 1.67i)5-s + (1.93 − 1.80i)7-s + (1.10 − 0.639i)11-s + (−2.52 − 1.45i)13-s − 0.475·17-s − 6.10i·19-s + (−6.51 − 3.75i)23-s + (0.635 + 1.09i)25-s + (−3.76 + 2.17i)29-s + (−4.21 − 2.43i)31-s + (−1.15 − 4.97i)35-s + 1.76·37-s + (−1.16 + 2.01i)41-s + (−4.63 − 8.03i)43-s + (4.00 + 6.93i)47-s + ⋯
L(s)  = 1  + (0.431 − 0.747i)5-s + (0.729 − 0.683i)7-s + (0.333 − 0.192i)11-s + (−0.701 − 0.404i)13-s − 0.115·17-s − 1.40i·19-s + (−1.35 − 0.783i)23-s + (0.127 + 0.219i)25-s + (−0.699 + 0.403i)29-s + (−0.757 − 0.437i)31-s + (−0.195 − 0.841i)35-s + 0.289·37-s + (−0.181 + 0.314i)41-s + (−0.707 − 1.22i)43-s + (0.584 + 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.739 + 0.673i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.739 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.516052994\)
\(L(\frac12)\) \(\approx\) \(1.516052994\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 + 1.80i)T \)
good5 \( 1 + (-0.965 + 1.67i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.10 + 0.639i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.52 + 1.45i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.475T + 17T^{2} \)
19 \( 1 + 6.10iT - 19T^{2} \)
23 \( 1 + (6.51 + 3.75i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.76 - 2.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.21 + 2.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.76T + 37T^{2} \)
41 \( 1 + (1.16 - 2.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.63 + 8.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.00 - 6.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + (-1.74 + 3.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.26 + 2.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.602 - 1.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.2iT - 71T^{2} \)
73 \( 1 - 1.84iT - 73T^{2} \)
79 \( 1 + (8.54 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.225 - 0.390i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-9.22 + 5.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.561689318657986238604959727786, −7.62172563278304025623071305241, −7.10280070333175712327154652273, −6.06611630733963505290146819723, −5.23757608685067257035992912266, −4.62252141413292140943521674604, −3.84549345609982287891293923313, −2.55706152265215003590530476400, −1.57853888494750700242197917576, −0.44419447774535751786364981687, 1.74334217106691711031310092336, 2.22196650530372673575101167926, 3.43669095949688993665808047779, 4.29895278529521442369407001093, 5.33657905152340515146520394953, 5.92793868613482212105071845672, 6.71711923721905285721926917721, 7.56234065396252154632264814787, 8.186764981014604857765804060119, 9.029215527377577776267514820545

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