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Properties

Label 2-3024-63.5-c1-0-32
Degree $2$
Conductor $3024$
Sign $0.985 - 0.168i$
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  + (0.905 + 1.56i)5-s + (2.60 − 0.449i)7-s + (−0.221 − 0.127i)11-s + (5.77 + 3.33i)13-s + (−1.99 − 3.46i)17-s + (−1.24 − 0.719i)19-s + (4.90 − 2.83i)23-s + (0.858 − 1.48i)25-s + (4.18 − 2.41i)29-s − 10.1i·31-s + (3.06 + 3.68i)35-s + (−1.65 + 2.86i)37-s + (−5.10 + 8.83i)41-s + (−1.12 − 1.94i)43-s + 11.9·47-s + ⋯
L(s)  = 1  + (0.405 + 0.701i)5-s + (0.985 − 0.169i)7-s + (−0.0668 − 0.0385i)11-s + (1.60 + 0.925i)13-s + (−0.484 − 0.839i)17-s + (−0.286 − 0.165i)19-s + (1.02 − 0.590i)23-s + (0.171 − 0.297i)25-s + (0.777 − 0.448i)29-s − 1.82i·31-s + (0.518 + 0.622i)35-s + (−0.272 + 0.471i)37-s + (−0.796 + 1.37i)41-s + (−0.171 − 0.296i)43-s + 1.74·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.985 - 0.168i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.985 - 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.528219168\)
\(L(\frac12)\) \(\approx\) \(2.528219168\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 + 0.449i)T \)
good5 \( 1 + (-0.905 - 1.56i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.221 + 0.127i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.77 - 3.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.99 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.24 + 0.719i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.90 + 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.18 + 2.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.1iT - 31T^{2} \)
37 \( 1 + (1.65 - 2.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.10 - 8.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.12 + 1.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + (3.97 - 2.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.11T + 59T^{2} \)
61 \( 1 - 9.93iT - 61T^{2} \)
67 \( 1 + 1.92T + 67T^{2} \)
71 \( 1 + 7.31iT - 71T^{2} \)
73 \( 1 + (2.47 - 1.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.66T + 79T^{2} \)
83 \( 1 + (2.68 + 4.64i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.378 - 0.655i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.21 - 2.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.734470573792210748124364450094, −8.072287993610643706803374933629, −7.12697183325802595243157102918, −6.49180738213283170518304826281, −5.86569969970818745700433109218, −4.69468777788470701341174494099, −4.20269514157720763003379127114, −2.97100010321508913012986873092, −2.13515177520923516342017244921, −1.01658843870894318074997283536, 1.11689995570378036129934372007, 1.73284406443960406663557964018, 3.11955227214388866550772351697, 3.99944151219541020958544465308, 5.07489366363693642111145724667, 5.43890440130658384822838808747, 6.34059977729132116852529242453, 7.24297625846651210487143735859, 8.238241297397968684100282204804, 8.690463124504868501501798021466

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