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Properties

Label 2-3024-63.41-c1-0-25
Degree $2$
Conductor $3024$
Sign $0.893 + 0.449i$
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  + (0.422 − 0.731i)5-s + (−2.10 − 1.59i)7-s + (−0.791 + 0.456i)11-s + (0.472 + 0.272i)13-s + 4.77·17-s + 3.15i·19-s + (1.39 + 0.804i)23-s + (2.14 + 3.71i)25-s + (−5.56 + 3.21i)29-s + (1.57 + 0.908i)31-s + (−2.05 + 0.869i)35-s + 7.14·37-s + (2.82 − 4.88i)41-s + (1.84 + 3.19i)43-s + (−6.75 − 11.6i)47-s + ⋯
L(s)  = 1  + (0.188 − 0.327i)5-s + (−0.797 − 0.603i)7-s + (−0.238 + 0.137i)11-s + (0.131 + 0.0756i)13-s + 1.15·17-s + 0.723i·19-s + (0.290 + 0.167i)23-s + (0.428 + 0.742i)25-s + (−1.03 + 0.596i)29-s + (0.282 + 0.163i)31-s + (−0.348 + 0.146i)35-s + 1.17·37-s + (0.440 − 0.763i)41-s + (0.281 + 0.487i)43-s + (−0.984 − 1.70i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.893 + 0.449i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.893 + 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.693824278\)
\(L(\frac12)\) \(\approx\) \(1.693824278\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 + 1.59i)T \)
good5 \( 1 + (-0.422 + 0.731i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.791 - 0.456i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.472 - 0.272i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.77T + 17T^{2} \)
19 \( 1 - 3.15iT - 19T^{2} \)
23 \( 1 + (-1.39 - 0.804i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.56 - 3.21i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.57 - 0.908i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.14T + 37T^{2} \)
41 \( 1 + (-2.82 + 4.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.84 - 3.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.75 + 11.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.49iT - 53T^{2} \)
59 \( 1 + (0.279 - 0.483i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.9 + 6.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.06 + 5.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.24iT - 71T^{2} \)
73 \( 1 + 8.87iT - 73T^{2} \)
79 \( 1 + (-5.58 - 9.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.122 + 0.211i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.19T + 89T^{2} \)
97 \( 1 + (-12.2 + 7.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.714942801748782021011416983473, −7.80629132454091336881149788837, −7.24975533539702277866165767236, −6.38734232569955158542790023114, −5.60163557617406669295976160831, −4.89026760475310679874653579244, −3.72562246804011194901543496718, −3.25106588302791435933648202727, −1.88655483785628026615856506373, −0.74505966077144733789808007031, 0.853418721423934991699509048905, 2.43119728605224137603531700298, 2.97137065474566374485305490355, 3.97266457042357543572034956449, 5.02382863707363135137513778232, 5.89232536566436095430743724670, 6.35504250910559188435398791126, 7.28430202867397140657264343075, 8.018007649786077901884466429995, 8.830472579295859467583807558822

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