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Properties

Label 1-3024-3024.2075-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.950 - 0.310i$
Analytic cond. $324.973$
Root an. cond. $324.973$
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 + (0.642 + 0.766i)13-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.642 + 0.766i)29-s + (0.766 − 0.642i)31-s + i·37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (−0.766 − 0.642i)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 + (0.642 + 0.766i)13-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.642 + 0.766i)29-s + (0.766 − 0.642i)31-s + i·37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (−0.766 − 0.642i)47-s + (−0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.950 - 0.310i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.950 - 0.310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2607793831 + 1.637652891i\)
\(L(\frac12)\) \(\approx\) \(-0.2607793831 + 1.637652891i\)
\(L(1)\) \(\approx\) \(0.9713305107 + 0.4861053260i\)
\(L(1)\) \(\approx\) \(0.9713305107 + 0.4861053260i\)

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 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.642 + 0.766i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.642 + 0.766i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−18.50303625131460524692989337091, −17.61311970186955463766420753382, −17.271451801454781890767535706821, −16.30805404258801861411009177416, −15.81986321172557647473473060296, −15.13021521414603508306157627285, −14.07167153112627280058101557351, −13.45665607291916117280128064805, −13.00471076436149922203003704837, −12.31889541205014024113478049543, −11.221254466079493297272367875062, −10.75977497341408325709497435032, −9.92289458415367024128194349673, −9.03266402319216862591949609301, −8.57895995276449079025837599819, −7.84622266810545882469378631512, −6.77825718081225925767868333491, −6.05780135778369244317640810827, −5.26645338399871470904698406419, −4.76351251139299546772183859716, −3.67412108389270489464444120104, −2.7435317659982219427735715599, −2.01460351774627535802540355028, −0.810268788586729541455823196261, −0.2945315662622700402782459005, 1.39809178209918767490290103736, 1.9875426375853525485551294777, 2.8581145972344647514841963471, 3.74968013308070939744449985833, 4.61741102941721699186748812968, 5.463391682580484318033682308922, 6.49623928007341539484904388804, 6.65850283131793761593925079228, 7.76628477212214068763233086906, 8.48912765903199635063469189776, 9.421959460478082907969079691008, 10.00227598098230867211305575946, 10.82924201645574609955714259870, 11.21652512721856844197723900217, 12.28042983298862375327426677527, 13.16945502488756502764823698467, 13.44703593024584014671015933114, 14.51630071171403608219804265278, 15.0018721022799102534137794799, 15.56847880859784556737057632995, 16.678369738177980755079161581358, 17.15574318934607323948227726811, 17.943661717669065133967035743827, 18.56600556629523993022503487310, 19.06395594091123288758313311090

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