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Properties

Label 1-2475-2475.1039-r0-0-0
Degree $1$
Conductor $2475$
Sign $-0.0596 - 0.998i$
Analytic cond. $11.4938$
Root an. cond. $11.4938$
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  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.669 − 0.743i)7-s + (0.809 + 0.587i)8-s + (0.104 − 0.994i)13-s + (−0.5 − 0.866i)14-s + (0.669 + 0.743i)16-s − 17-s + (−0.809 − 0.587i)19-s + (0.978 − 0.207i)23-s + (0.309 − 0.951i)26-s + (−0.309 − 0.951i)28-s + (−0.104 − 0.994i)29-s + (−0.978 + 0.207i)31-s + (0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.669 − 0.743i)7-s + (0.809 + 0.587i)8-s + (0.104 − 0.994i)13-s + (−0.5 − 0.866i)14-s + (0.669 + 0.743i)16-s − 17-s + (−0.809 − 0.587i)19-s + (0.978 − 0.207i)23-s + (0.309 − 0.951i)26-s + (−0.309 − 0.951i)28-s + (−0.104 − 0.994i)29-s + (−0.978 + 0.207i)31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0596 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0596 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.0596 - 0.998i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ -0.0596 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.381053818 - 1.466098269i\)
\(L(\frac12)\) \(\approx\) \(1.381053818 - 1.466098269i\)
\(L(1)\) \(\approx\) \(1.571026332 - 0.1831839337i\)
\(L(1)\) \(\approx\) \(1.571026332 - 0.1831839337i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 - T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.669 - 0.743i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−19.53147373765287660791193209137, −19.23730148010615921536245708832, −18.468681524831593432659189247609, −17.47904711742133388205997905352, −16.35957184904498204533617904641, −16.187323151095214042554816638043, −15.15268005605725067910919797409, −14.730257094505843115995114388536, −13.87370387977802958314223254770, −13.07842286920574211996378500014, −12.585755994782126292936379582981, −11.894496654055766435948585857797, −11.033391933521765331895655492892, −10.560812694202859400468498191015, −9.28533640330004000430078134654, −9.01420883541862029064644702151, −7.70489557723382429505533495794, −6.75492512590873122241904616009, −6.33219284921427887477121989834, −5.47916188621114569348641685283, −4.650465813460724773510863628879, −3.872731500216525845768299071244, −3.04426834332967649679541883084, −2.21907870038617841721585865388, −1.46304301051577859129852533622, 0.40477578395734477937798104982, 1.82060419917467060696732833466, 2.77727721723050918470960769702, 3.48602730403073981264886811578, 4.28375522640264126690196538358, 5.01131855524073076699401369054, 5.93540023808232510740215941050, 6.675709836892185973858605300586, 7.217757472628690618459995871669, 8.10021596487537353546603654445, 8.984518067668472083982972725902, 10.02522173203001722179437187756, 10.917098340094673284318724550339, 11.1485400120548188507175686142, 12.48670879655399452334558794371, 12.88785868997117929097108493085, 13.45662135562947854027959480442, 14.18521288878435582034353694458, 15.10176842359702155809554783439, 15.57830526978930521165932881304, 16.2508284460261591382332293498, 17.278026070632477253929655879291, 17.38197858449619752055400017985, 18.72870175326242817090229835997, 19.59535845380330451920884181308

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