Properties

Degree $2$
Conductor $3024$
Sign $0.755 + 0.654i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·5-s + (2 + 1.73i)7-s − 3.46i·11-s + 3.46i·13-s − 1.73i·17-s − 2·19-s + 2.00·25-s + 6·29-s + 2·31-s + (2.99 − 3.46i)35-s + 37-s − 5.19i·41-s − 1.73i·43-s − 3·47-s + (1.00 + 6.92i)49-s + ⋯
L(s)  = 1  − 0.774i·5-s + (0.755 + 0.654i)7-s − 1.04i·11-s + 0.960i·13-s − 0.420i·17-s − 0.458·19-s + 0.400·25-s + 1.11·29-s + 0.359·31-s + (0.507 − 0.585i)35-s + 0.164·37-s − 0.811i·41-s − 0.264i·43-s − 0.437·47-s + (0.142 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.755 + 0.654i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.011822455\)
\(L(\frac12)\) \(\approx\) \(2.011822455\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 1.73iT - 79T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−8.615569725576451338330027133995, −8.213700704646330522583304468171, −7.12082006532309916508943525772, −6.30170367510470301474186080147, −5.47163910060977024061727019801, −4.81809910941579823533122901722, −4.07162418257711142966903336517, −2.87684242128108222421340472720, −1.89020282111464465092488994864, −0.76973689203848763629691502914, 1.05047054358958422598057850152, 2.24569575228573954634634820170, 3.12895080752269676522959116115, 4.20492015467822639094068765075, 4.80627462284735285995418671014, 5.79484833374628755163853077930, 6.73961271961020402772990681626, 7.24614113952464245359521577162, 8.034925581622747721991614425366, 8.596145739783728379344149036936

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