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Properties

Label 1-3168-3168.493-r0-0-0
Degree $1$
Conductor $3168$
Sign $0.396 - 0.917i$
Analytic cond. $14.7121$
Root an. cond. $14.7121$
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  + (−0.777 − 0.629i)5-s + (−0.743 − 0.669i)7-s + (0.933 + 0.358i)13-s + (0.809 − 0.587i)17-s + (0.891 − 0.453i)19-s + (−0.866 − 0.5i)23-s + (0.207 + 0.978i)25-s + (−0.998 + 0.0523i)29-s + (0.913 − 0.406i)31-s + (0.156 + 0.987i)35-s + (0.891 + 0.453i)37-s + (−0.743 + 0.669i)41-s + (0.965 + 0.258i)43-s + (0.978 − 0.207i)47-s + (0.104 + 0.994i)49-s + ⋯
L(s)  = 1  + (−0.777 − 0.629i)5-s + (−0.743 − 0.669i)7-s + (0.933 + 0.358i)13-s + (0.809 − 0.587i)17-s + (0.891 − 0.453i)19-s + (−0.866 − 0.5i)23-s + (0.207 + 0.978i)25-s + (−0.998 + 0.0523i)29-s + (0.913 − 0.406i)31-s + (0.156 + 0.987i)35-s + (0.891 + 0.453i)37-s + (−0.743 + 0.669i)41-s + (0.965 + 0.258i)43-s + (0.978 − 0.207i)47-s + (0.104 + 0.994i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3168\)    =    \(2^{5} \cdot 3^{2} \cdot 11\)
Sign: $0.396 - 0.917i$
Analytic conductor: \(14.7121\)
Root analytic conductor: \(14.7121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3168} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3168,\ (0:\ ),\ 0.396 - 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134651754 - 0.7457261989i\)
\(L(\frac12)\) \(\approx\) \(1.134651754 - 0.7457261989i\)
\(L(1)\) \(\approx\) \(0.9214963033 - 0.2247943414i\)
\(L(1)\) \(\approx\) \(0.9214963033 - 0.2247943414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.777 - 0.629i)T \)
7 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.933 + 0.358i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.891 - 0.453i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.998 + 0.0523i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.891 + 0.453i)T \)
41 \( 1 + (-0.743 + 0.669i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (0.978 - 0.207i)T \)
53 \( 1 + (0.987 - 0.156i)T \)
59 \( 1 + (-0.838 + 0.544i)T \)
61 \( 1 + (-0.358 - 0.933i)T \)
67 \( 1 + (0.965 - 0.258i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.358 + 0.933i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−19.02567131933403138673923122826, −18.46077216181648578206188245171, −17.96625611850645308047905124442, −16.86211693650380727534542378579, −16.13643975491180082308572225041, −15.58824610755902867551513698102, −15.113248835238339707607431361849, −14.19029477130696664301118743198, −13.57745838791911260404848934283, −12.57344433432988708770691848929, −12.10291868869818744040282287133, −11.40827180335012778618657588218, −10.57556089591076571571785919160, −9.96057850124214384022248254248, −9.12615863738708073394395520331, −8.26869343601309900447642828168, −7.67442697502617776675889763893, −6.90396830247904980044625271495, −5.79732207448251893693439994736, −5.75806409902423315936144534137, −4.22969836860312033604020779246, −3.53955229322817828014357972318, −3.05080534412075347601285760310, −2.01313968607170299081049202013, −0.81696281093882909038467610443, 0.61510877338437935930813586500, 1.25257433528913509307065683013, 2.64836905682801747403220249500, 3.54368158576388413619113839890, 4.06355576072159101143728706720, 4.88596518554338686603627093530, 5.82828948984059385604669538278, 6.59484283080516131162484219032, 7.504266739281222133442398337855, 7.94347013206272675256904525267, 8.9271046321501626273894108563, 9.54036544614509072556208733128, 10.274153663661037685835540194883, 11.22452124443628397309906363828, 11.771950173802349949418531878697, 12.48399362706470005122742351834, 13.32612725210768643729039619216, 13.73703955147101845898983169138, 14.63600749042338827178377842011, 15.66531610137558617745522908912, 16.00916420119625534755696518287, 16.66649036126918521265983644257, 17.18631395987554846679219967250, 18.448095436377249251443486411470, 18.68676791344359473000188176192

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