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Properties

Label 1-4800-4800.2411-r0-0-0
Degree $1$
Conductor $4800$
Sign $0.947 - 0.320i$
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 + (0.649 − 0.760i)11-s + (−0.760 + 0.649i)13-s + (−0.587 − 0.809i)17-s + (−0.852 − 0.522i)19-s + (0.453 + 0.891i)23-s + (0.972 + 0.233i)29-s + (−0.809 + 0.587i)31-s + (−0.996 − 0.0784i)37-s + (0.891 + 0.453i)41-s + (0.923 + 0.382i)43-s + (−0.587 + 0.809i)47-s + i·49-s + (0.522 + 0.852i)53-s + (0.996 + 0.0784i)59-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)7-s + (0.649 − 0.760i)11-s + (−0.760 + 0.649i)13-s + (−0.587 − 0.809i)17-s + (−0.852 − 0.522i)19-s + (0.453 + 0.891i)23-s + (0.972 + 0.233i)29-s + (−0.809 + 0.587i)31-s + (−0.996 − 0.0784i)37-s + (0.891 + 0.453i)41-s + (0.923 + 0.382i)43-s + (−0.587 + 0.809i)47-s + i·49-s + (0.522 + 0.852i)53-s + (0.996 + 0.0784i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(22.2911\)
Root analytic conductor: \(22.2911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (2411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4800,\ (0:\ ),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.263349439 - 0.2077305292i\)
\(L(\frac12)\) \(\approx\) \(1.263349439 - 0.2077305292i\)
\(L(1)\) \(\approx\) \(0.9255924476 - 0.08715400769i\)
\(L(1)\) \(\approx\) \(0.9255924476 - 0.08715400769i\)

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 \( 1 + (0.649 - 0.760i)T \)
13 \( 1 + (-0.760 + 0.649i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.852 - 0.522i)T \)
23 \( 1 + (0.453 + 0.891i)T \)
29 \( 1 + (0.972 + 0.233i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.996 - 0.0784i)T \)
41 \( 1 + (0.891 + 0.453i)T \)
43 \( 1 + (0.923 + 0.382i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (0.522 + 0.852i)T \)
59 \( 1 + (0.996 + 0.0784i)T \)
61 \( 1 + (0.0784 + 0.996i)T \)
67 \( 1 + (-0.522 + 0.852i)T \)
71 \( 1 + (-0.156 - 0.987i)T \)
73 \( 1 + (-0.453 - 0.891i)T \)
79 \( 1 + (-0.587 + 0.809i)T \)
83 \( 1 + (0.852 + 0.522i)T \)
89 \( 1 + (0.891 - 0.453i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−18.10640146411780042261669518824, −17.3962243614409047851348304755, −16.99742545207516026356912862406, −16.07967558810371521944889836545, −15.4713576010944900633510009853, −14.72187884143427067346979328383, −14.500308977892350278821242421931, −13.21436346281345523407609129016, −12.70994423147796568188976907929, −12.2858208961174880125270266104, −11.55352137499073762152221963652, −10.44921784154906348191950748744, −10.180859557190237422491008096956, −9.21494100151845151700834702700, −8.73165865479896545043018387214, −7.951261719156911133050772905911, −6.98578825524709678301780069812, −6.48188881466441140600755607181, −5.75198206457662507366199952812, −4.92666741318122888116300522152, −4.123348064074555711997086553947, −3.41236465465239848764067051224, −2.32632840344571768073337481421, −2.01507535588946463498031841320, −0.575246973091440726800054863654, 0.58299116179123090677392633010, 1.50661416601853285218339019465, 2.61781007028167424928310714216, 3.22780129764454975845149446670, 4.15883909974146997120689144875, 4.66748416988190164317240547597, 5.67799674114323476577855437367, 6.518006337314539515992097721460, 7.01297510796983152129079985890, 7.604734307200903447058930115343, 8.86506196397022462133230356598, 9.08606991387733068954710978419, 9.91002794697653634071447481433, 10.72513076307721331506470766578, 11.27927088658583657213516650681, 12.02444207282942289172849404537, 12.78058095024082925586213396267, 13.474686389107711476487281268817, 14.03870258886727533348511600190, 14.61930813099247428239394014297, 15.53300259761372659511005277153, 16.28334565674125042069725403766, 16.59416048167056427332878710931, 17.51061434917565237976259411602, 17.84677013447465306930570259472

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