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Properties

Label 1-3024-3024.1363-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.947 - 0.318i$
Analytic cond. $14.0433$
Root an. cond. $14.0433$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 + 0.766i)5-s + (−0.642 + 0.766i)11-s + (−0.342 + 0.939i)13-s − 17-s − i·19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.342 − 0.939i)29-s + (0.173 + 0.984i)31-s + (0.866 + 0.5i)37-s + (−0.939 − 0.342i)41-s + (0.984 + 0.173i)43-s + (0.173 − 0.984i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)5-s + (−0.642 + 0.766i)11-s + (−0.342 + 0.939i)13-s − 17-s − i·19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.342 − 0.939i)29-s + (0.173 + 0.984i)31-s + (0.866 + 0.5i)37-s + (−0.939 − 0.342i)41-s + (0.984 + 0.173i)43-s + (0.173 − 0.984i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.947 - 0.318i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.947 - 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09073033710 + 0.5549355845i\)
\(L(\frac12)\) \(\approx\) \(-0.09073033710 + 0.5549355845i\)
\(L(1)\) \(\approx\) \(0.8531295863 + 0.3032329603i\)
\(L(1)\) \(\approx\) \(0.8531295863 + 0.3032329603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (-0.642 + 0.766i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 - T \)
19 \( 1 - iT \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.342 - 0.939i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (-0.342 + 0.939i)T \)
61 \( 1 + (-0.984 - 0.173i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.342 - 0.939i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.52276799778594012228261456629, −17.83478183417570062259030636916, −17.40616161968105468155018272968, −16.559760608636237833589806259675, −15.882785287212111416096583716257, −15.33748378861309060605153096843, −14.37045362331476351293322729778, −13.537134804966844063078061308316, −13.11367133229774905250193456276, −12.54216282828560191500936299122, −11.507134444713836803546010760003, −10.86078680104127430747519209398, −10.072879285746615891320607231661, −9.34513183164363716530549212153, −8.647542229864655625741161547344, −7.99387713497099714310323333753, −7.151620217529673840721412530141, −6.07455154735998591620503260890, −5.588993142648098201371087846896, −4.82545013673320536682755049160, −4.03635968447000453798539902469, −2.84227706144190288138154502312, −2.29157526031931736390436592695, −1.14492717505600012477462202256, −0.16190205175286022341586142040, 1.69942724066463248564154639755, 2.15816237952932816567605173987, 2.989910115225261211013526973079, 4.103067596501221247312377968221, 4.71842685662736282659664154498, 5.80040498106063510777540625902, 6.36657206582517806474439137234, 7.16623308012592143140210316623, 7.78076147365774057698388389754, 8.822826624498904789141562589918, 9.5587549637162439860288850945, 10.283740980227695856135252065414, 10.68928480021642335743939271096, 11.79404350132880300063424578979, 12.23713085775549071502669578721, 13.37904659807703955706022268512, 13.70881662797063606315254822228, 14.62086707516023843480881073722, 15.088561109565650913257052475233, 15.917761400396422593397057824872, 16.73276625905435911093094773354, 17.4189028849164977486644490519, 18.17017362546262690818697459612, 18.52900863261483125202653076491, 19.36771946004262093439786249723

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