Properties

Label 2-3024-63.4-c1-0-25
Degree $2$
Conductor $3024$
Sign $0.999 + 0.00294i$
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  − 1.76·5-s + (1.85 + 1.88i)7-s + 6.12·11-s + (−0.380 + 0.658i)13-s + (3.42 − 5.92i)17-s + (−0.971 − 1.68i)19-s − 0.421·23-s − 1.89·25-s + (−0.732 − 1.26i)29-s + (3.85 + 6.67i)31-s + (−3.26 − 3.32i)35-s + (1.44 + 2.49i)37-s + (3.47 − 6.01i)41-s + (−4.33 − 7.49i)43-s + (−0.830 + 1.43i)47-s + ⋯
L(s)  = 1  − 0.787·5-s + (0.699 + 0.714i)7-s + 1.84·11-s + (−0.105 + 0.182i)13-s + (0.829 − 1.43i)17-s + (−0.222 − 0.385i)19-s − 0.0877·23-s − 0.379·25-s + (−0.135 − 0.235i)29-s + (0.691 + 1.19i)31-s + (−0.551 − 0.562i)35-s + (0.237 + 0.410i)37-s + (0.542 − 0.939i)41-s + (−0.660 − 1.14i)43-s + (−0.121 + 0.209i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.00294i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.999 + 0.00294i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.964625082\)
\(L(\frac12)\) \(\approx\) \(1.964625082\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.85 - 1.88i)T \)
good5 \( 1 + 1.76T + 5T^{2} \)
11 \( 1 - 6.12T + 11T^{2} \)
13 \( 1 + (0.380 - 0.658i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.42 + 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.971 + 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.421T + 23T^{2} \)
29 \( 1 + (0.732 + 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.47 + 6.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.33 + 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.830 - 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.112 + 0.195i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.993 + 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.17 + 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.39 - 5.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (-0.153 + 0.265i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.72 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.56 + 2.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.81 + 3.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.714297036974460374303484465752, −8.054630136299510439554257990924, −7.17522018303721267968357245676, −6.62445251261970066351018317831, −5.58149192285459349391795291836, −4.80243154287378268862938324661, −4.02099081526398689432690925668, −3.18626104204633165493883524305, −2.01492531660253585232327081576, −0.862182705804347412208479469874, 0.954338697660650758644828365425, 1.81208602696050159770034749459, 3.42875879472191157855926364381, 4.03128301947096485354020855155, 4.51664047719054178576162137638, 5.83016712084265005356912335173, 6.44041711485531466409360826961, 7.36565794902752576639873492405, 8.023732401312006071232191294585, 8.466409850803858274359039688595

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