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Properties

Label 1-3168-3168.2819-r0-0-0
Degree $1$
Conductor $3168$
Sign $-0.331 + 0.943i$
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  + (−0.838 + 0.544i)5-s + (−0.994 − 0.104i)7-s + (0.0523 − 0.998i)13-s + (0.309 + 0.951i)17-s + (−0.987 − 0.156i)19-s + (−0.866 − 0.5i)23-s + (0.406 − 0.913i)25-s + (0.777 − 0.629i)29-s + (−0.669 − 0.743i)31-s + (0.891 − 0.453i)35-s + (0.987 − 0.156i)37-s + (−0.994 + 0.104i)41-s + (0.965 + 0.258i)43-s + (−0.913 − 0.406i)47-s + (0.978 + 0.207i)49-s + ⋯
L(s)  = 1  + (−0.838 + 0.544i)5-s + (−0.994 − 0.104i)7-s + (0.0523 − 0.998i)13-s + (0.309 + 0.951i)17-s + (−0.987 − 0.156i)19-s + (−0.866 − 0.5i)23-s + (0.406 − 0.913i)25-s + (0.777 − 0.629i)29-s + (−0.669 − 0.743i)31-s + (0.891 − 0.453i)35-s + (0.987 − 0.156i)37-s + (−0.994 + 0.104i)41-s + (0.965 + 0.258i)43-s + (−0.913 − 0.406i)47-s + (0.978 + 0.207i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3168\)    =    \(2^{5} \cdot 3^{2} \cdot 11\)
Sign: $-0.331 + 0.943i$
Analytic conductor: \(14.7121\)
Root analytic conductor: \(14.7121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3168} (2819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3168,\ (0:\ ),\ -0.331 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2743619918 + 0.3874082655i\)
\(L(\frac12)\) \(\approx\) \(0.2743619918 + 0.3874082655i\)
\(L(1)\) \(\approx\) \(0.6999686521 + 0.04424174568i\)
\(L(1)\) \(\approx\) \(0.6999686521 + 0.04424174568i\)

Euler product

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

−18.83865331168385285707697663364, −18.23281569894235441637791482497, −17.16132229793414829699524616580, −16.4765357659748816283870120213, −16.03025998583195224885479246389, −15.52844246357717265746519863886, −14.484881141238685465269047547706, −13.92346006343711239677869384938, −12.88866846248895639987953344017, −12.53709091577375055395981864289, −11.70318456535797041097516600106, −11.19737440477865216890276256398, −10.08674443906352919842325858508, −9.50165285188202193209464483103, −8.73551449586262424197845878371, −8.13598195254108273807542456332, −7.08614916446610487369189616557, −6.68414131620033684346423014037, −5.66245625517519240500605583348, −4.822073241222551595841714207253, −4.01565147182820899738070109353, −3.41485771656659481502672037668, −2.44243277637341170839548473576, −1.37376834070720654496906884732, −0.19592468922267339483660000689, 0.79101433073229900342544732520, 2.279147916703580028869377372445, 2.946446191108680451444277117048, 3.85397524979749776883352052982, 4.23800282084093854123076293204, 5.56253238941022854424286370367, 6.30176815525758915118094524772, 6.81916820433397045004529732464, 7.948723544096168398107722723859, 8.151540653115652111707124098178, 9.26050132612481351234123861361, 10.217176640805904549127751223953, 10.52333102152869431107806981808, 11.38425781215950539609603100266, 12.299163619432834212556525585148, 12.74422120656805298994058321733, 13.46536861286041802776124586276, 14.45117528990348472330317518976, 15.12235377901458399231961993986, 15.55990922082551262425243262427, 16.37875871251907573832640853866, 16.963599315463275749728895508621, 17.86595805615380085730211343298, 18.60876581912515182816903092762, 19.17625488687615147034927927690

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