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Properties

Label 1-4800-4800.1211-r0-0-0
Degree $1$
Conductor $4800$
Sign $-0.443 + 0.896i$
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 + (0.760 + 0.649i)11-s + (0.649 + 0.760i)13-s + (−0.587 − 0.809i)17-s + (−0.522 + 0.852i)19-s + (−0.453 − 0.891i)23-s + (0.233 − 0.972i)29-s + (−0.809 + 0.587i)31-s + (0.0784 − 0.996i)37-s + (−0.891 − 0.453i)41-s + (−0.382 + 0.923i)43-s + (−0.587 + 0.809i)47-s + i·49-s + (−0.852 + 0.522i)53-s + (−0.0784 + 0.996i)59-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)7-s + (0.760 + 0.649i)11-s + (0.649 + 0.760i)13-s + (−0.587 − 0.809i)17-s + (−0.522 + 0.852i)19-s + (−0.453 − 0.891i)23-s + (0.233 − 0.972i)29-s + (−0.809 + 0.587i)31-s + (0.0784 − 0.996i)37-s + (−0.891 − 0.453i)41-s + (−0.382 + 0.923i)43-s + (−0.587 + 0.809i)47-s + i·49-s + (−0.852 + 0.522i)53-s + (−0.0784 + 0.996i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $-0.443 + 0.896i$
Analytic conductor: \(22.2911\)
Root analytic conductor: \(22.2911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (1211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4800,\ (0:\ ),\ -0.443 + 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7885494675 + 1.269943440i\)
\(L(\frac12)\) \(\approx\) \(0.7885494675 + 1.269943440i\)
\(L(1)\) \(\approx\) \(1.070431889 + 0.2794215819i\)
\(L(1)\) \(\approx\) \(1.070431889 + 0.2794215819i\)

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

−17.90092777611915911078542625106, −17.13286423985742671072653758684, −16.83572233911948215564526054652, −15.851919575895725057146297435424, −15.19885568943547759117548457249, −14.61689880413408476867621185699, −13.79443334742433347162010385381, −13.33020307143488830821824467655, −12.675145753593600653635709441241, −11.56494752311857242334094747358, −11.22105800129628351176207742037, −10.573096265731469801717118178540, −9.85652329732751660428087738904, −8.831366380753461627823414756631, −8.4065985055560955845070789766, −7.71496059262105216154899947519, −6.76382442410280262015067703767, −6.29395059903490800595766909502, −5.31192081471167903070449497921, −4.67951273102250628993624972317, −3.64442052940269410215922378337, −3.42456492030523802595018868385, −1.9954140400125966124246425239, −1.42207494443871813182815708251, −0.392139839967762312003601887710, 1.21215658253750747191091572398, 1.96072774610009749788892813898, 2.541924172936906842919321348873, 3.78978027254138867728051239264, 4.34240250738838689073157664867, 5.04612029466380451619434261269, 5.98998983393789637757276521208, 6.55150609808813637045897298952, 7.32013437683137164376048215526, 8.248926498855719704823332242900, 8.756002222918928509904217629091, 9.42803448466820819102450598658, 10.14689630963303183235008911481, 11.16372503677387834369710642183, 11.50968100088621022450256600689, 12.29291409260641333021364445807, 12.8036985810897979819690554734, 13.87586389713414440191811697241, 14.381234356032650900729455620026, 14.85584205559361847733815156668, 15.77798455826866201780413013597, 16.22981835869692740066947100755, 17.10727324366426844100483740172, 17.70403174559006286611561070796, 18.40078826307723156367115464996

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