Properties

Label 1-42e2-1764.1375-r0-0-0
Degree $1$
Conductor $1764$
Sign $0.763 + 0.645i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
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  + (0.988 + 0.149i)5-s + (0.733 + 0.680i)11-s + (0.733 + 0.680i)13-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.826 + 0.563i)29-s + 31-s + (0.0747 − 0.997i)37-s + (−0.365 − 0.930i)41-s + (−0.365 + 0.930i)43-s + (−0.222 + 0.974i)47-s + (0.0747 + 0.997i)53-s + (0.623 + 0.781i)55-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)5-s + (0.733 + 0.680i)11-s + (0.733 + 0.680i)13-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.826 + 0.563i)29-s + 31-s + (0.0747 − 0.997i)37-s + (−0.365 − 0.930i)41-s + (−0.365 + 0.930i)43-s + (−0.222 + 0.974i)47-s + (0.0747 + 0.997i)53-s + (0.623 + 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.763 + 0.645i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1375, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.763 + 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.981382723 + 0.7251631377i\)
\(L(\frac12)\) \(\approx\) \(1.981382723 + 0.7251631377i\)
\(L(1)\) \(\approx\) \(1.347773781 + 0.1887567268i\)
\(L(1)\) \(\approx\) \(1.347773781 + 0.1887567268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.988 + 0.149i)T \)
11 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (0.733 + 0.680i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + T \)
37 \( 1 + (0.0747 - 0.997i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (-0.365 + 0.930i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
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

−20.10071817570950616588409072768, −19.51131326630270939073926699994, −18.65794472314957102447771484157, −17.70855971766590443839749356282, −17.3918380248279979492727548495, −16.61722243876504423928483933632, −15.63491420636351297105585702736, −15.07121451585743789827448842673, −14.01655362463941259147816651920, −13.4020850127192342854665623644, −13.02795199392758305402257872494, −11.79888610319101253687662377753, −11.17152238697212202329810610883, −10.27152627962263830949996960325, −9.650883378689710144887045681523, −8.54497077386031176942024026763, −8.403486725128102332206057138721, −6.804797447115283538929805568787, −6.35990009924237997350677026385, −5.55980698473837942901332335415, −4.67189025878298908418049372885, −3.65763872260493001644681545495, −2.744296985055288905381565950585, −1.749573566945215620918743341085, −0.83697283591460067894291852638, 1.18020752133632321588459510460, 1.9983382933245921188790338033, 2.80970450095276690484377072442, 4.08148955735187962643157889245, 4.675700452385874531162823721071, 5.82601867656203035073062212487, 6.53347574538794621453447782395, 7.01234077401678569213891683715, 8.36209982479519440271837697899, 8.99630858603835411575377395322, 9.73460777148572896664398045352, 10.48567586088579494491292560574, 11.23894084945747582400053446509, 12.21176489375647036775827346937, 12.83687069347821557676853774762, 13.85800516430942827998957563966, 14.20248283111904371039588442426, 15.017587087896631703963558228536, 16.02130979699815797754264342029, 16.673903132767867154123134593608, 17.478300655788045541690830729730, 18.044097587062680948420444221677, 18.74509398447496122554584454270, 19.591924311196610671514681841821, 20.47559581859861736831170107508

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