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Properties

Label 1-4800-4800.1517-r0-0-0
Degree $1$
Conductor $4800$
Sign $-0.267 - 0.963i$
Analytic cond. $22.2911$
Root an. cond. $22.2911$
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  + (−0.707 − 0.707i)7-s + (0.852 + 0.522i)11-s + (0.852 − 0.522i)13-s + (0.309 + 0.951i)17-s + (0.0784 − 0.996i)19-s + (0.156 − 0.987i)23-s + (−0.760 − 0.649i)29-s + (−0.309 − 0.951i)31-s + (0.233 − 0.972i)37-s + (0.156 + 0.987i)41-s + (−0.923 + 0.382i)43-s + (0.309 − 0.951i)47-s − i·49-s + (0.0784 + 0.996i)53-s + (0.972 + 0.233i)59-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)7-s + (0.852 + 0.522i)11-s + (0.852 − 0.522i)13-s + (0.309 + 0.951i)17-s + (0.0784 − 0.996i)19-s + (0.156 − 0.987i)23-s + (−0.760 − 0.649i)29-s + (−0.309 − 0.951i)31-s + (0.233 − 0.972i)37-s + (0.156 + 0.987i)41-s + (−0.923 + 0.382i)43-s + (0.309 − 0.951i)47-s − i·49-s + (0.0784 + 0.996i)53-s + (0.972 + 0.233i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $-0.267 - 0.963i$
Analytic conductor: \(22.2911\)
Root analytic conductor: \(22.2911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (1517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4800,\ (0:\ ),\ -0.267 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8457446979 - 1.112714492i\)
\(L(\frac12)\) \(\approx\) \(0.8457446979 - 1.112714492i\)
\(L(1)\) \(\approx\) \(1.005119947 - 0.2260204740i\)
\(L(1)\) \(\approx\) \(1.005119947 - 0.2260204740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.852 + 0.522i)T \)
13 \( 1 + (0.852 - 0.522i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.0784 - 0.996i)T \)
23 \( 1 + (0.156 - 0.987i)T \)
29 \( 1 + (-0.760 - 0.649i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.233 - 0.972i)T \)
41 \( 1 + (0.156 + 0.987i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.0784 + 0.996i)T \)
59 \( 1 + (0.972 + 0.233i)T \)
61 \( 1 + (-0.233 - 0.972i)T \)
67 \( 1 + (0.0784 - 0.996i)T \)
71 \( 1 + (-0.453 - 0.891i)T \)
73 \( 1 + (0.156 - 0.987i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (-0.996 - 0.0784i)T \)
89 \( 1 + (-0.156 + 0.987i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−18.49934029400644698014071623505, −17.728109976455629819320050028261, −16.826383619856403067182847140566, −16.21737928077627929171478501568, −15.90238056320433667954815031079, −14.945560880545645608727081911388, −14.28635216531081993433122059692, −13.664455506350153088597871601243, −12.99056883062622773805084776655, −12.14215797317342007043118447003, −11.66380218864092687525580610584, −11.04527453736458649286984153720, −10.02262064364569400536281564825, −9.45091364417848469462583225249, −8.81223226750444311791811750155, −8.26968678628551393241811983971, −7.08854295329839757846250524510, −6.7096184090590343921212893925, −5.66247215997394658477119028668, −5.47182460748509581048694178536, −4.1508765136329635974600337249, −3.492792579980752859199260226124, −2.94639321307334179299170935226, −1.773770323788187762899530313838, −1.117743367697354023729279498948, 0.40938522402515880935469649260, 1.289159022568508373715519174992, 2.24773775929394036182778043574, 3.2305014186415882263747183932, 3.923908369158670952930688363307, 4.42641788892262035261307535219, 5.54281610900050025927735794824, 6.32440969534197493660060545208, 6.7575700046721719960096542958, 7.64512654962208425858972838467, 8.30034585155459563270481903182, 9.26523936590765866332140841252, 9.646266593663995787763786037064, 10.63009147452000164114054816840, 10.97205720365038883784371143500, 11.92315456533319081922056583269, 12.68154524314650449594890254624, 13.24364788054092920985542369447, 13.74460235893780394717456476670, 14.82902313292950743420628545124, 15.07226987945399422009122000877, 16.01907168239688001470244011752, 16.74828151727372725842865588733, 17.08179456175532415277351284223, 17.93349864262210992164670169824

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