Properties

Label 1-1380-1380.59-r0-0-0
Degree $1$
Conductor $1380$
Sign $0.747 + 0.663i$
Analytic cond. $6.40869$
Root an. cond. $6.40869$
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  + (0.841 + 0.540i)7-s + (−0.142 − 0.989i)11-s + (−0.841 + 0.540i)13-s + (−0.959 + 0.281i)17-s + (0.959 + 0.281i)19-s + (0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.654 − 0.755i)37-s + (0.654 + 0.755i)41-s + (0.415 + 0.909i)43-s − 47-s + (0.415 + 0.909i)49-s + (0.841 + 0.540i)53-s + (0.841 − 0.540i)59-s + (0.415 − 0.909i)61-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)7-s + (−0.142 − 0.989i)11-s + (−0.841 + 0.540i)13-s + (−0.959 + 0.281i)17-s + (0.959 + 0.281i)19-s + (0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.654 − 0.755i)37-s + (0.654 + 0.755i)41-s + (0.415 + 0.909i)43-s − 47-s + (0.415 + 0.909i)49-s + (0.841 + 0.540i)53-s + (0.841 − 0.540i)59-s + (0.415 − 0.909i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.747 + 0.663i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ 0.747 + 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.464326774 + 0.5560091416i\)
\(L(\frac12)\) \(\approx\) \(1.464326774 + 0.5560091416i\)
\(L(1)\) \(\approx\) \(1.123942303 + 0.1297069025i\)
\(L(1)\) \(\approx\) \(1.123942303 + 0.1297069025i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (0.841 + 0.540i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 + (0.654 - 0.755i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 - T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (0.654 + 0.755i)T \)
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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

−20.5627001814755142654480553416, −20.112337615656319626118320437057, −19.44820409163343121973990334387, −18.11869006862072587636987546002, −17.84384367906111236187739115661, −17.12638032448682192116055356590, −16.20724056457077410776478062513, −15.24730375597789660309858572024, −14.78250776774680504500477264396, −13.84324648916732091874745336638, −13.17202873402472129234273855401, −12.20673721800520178838541820896, −11.5402560071281208896212171605, −10.64633946637093755218158885995, −9.9366979240987334390685234464, −9.12255663211141596627172440427, −8.0492205324925726350479904638, −7.396392498185767308331215166085, −6.75364988582313723924792617760, −5.39864934673723919562796769241, −4.78454753463127674314578865135, −4.02685185729612506223563047999, −2.72503272915542821387380366999, −1.94495478628160789563910514530, −0.70209974543402786641253281611, 1.05604957880900602473856222455, 2.17177249308802298015816960938, 2.96718120944511351719082208394, 4.17706446966292355772488969625, 4.986337043881921790060753066172, 5.77047733591322907310005101529, 6.69450749708106111098998461870, 7.68927049270343272912306597874, 8.424875633756769439358536147290, 9.14405277725961271661045366736, 10.03640686382846531316393152044, 11.13627804095567710767863840867, 11.51200511645533505226849554395, 12.42079072875592999410498196316, 13.28592627138476475386678783234, 14.31149202580440880183447887412, 14.558190679168968152533816077, 15.767693800999620435859260803619, 16.215378813408806040159800359023, 17.26701376626258000742424432156, 17.9226542520259772624168742186, 18.57133629847951225938827611837, 19.49247753944898985938361705611, 20.02659402910965265726727003655, 21.19843340547460485069970564805

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