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Properties

Label 1-3168-3168.2149-r0-0-0
Degree $1$
Conductor $3168$
Sign $-0.163 - 0.986i$
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  + (−0.0523 + 0.998i)5-s + (−0.406 + 0.913i)7-s + (−0.838 + 0.544i)13-s + (−0.309 + 0.951i)17-s + (−0.987 + 0.156i)19-s + (0.866 + 0.5i)23-s + (−0.994 − 0.104i)25-s + (−0.933 + 0.358i)29-s + (−0.978 − 0.207i)31-s + (−0.891 − 0.453i)35-s + (−0.987 − 0.156i)37-s + (−0.406 − 0.913i)41-s + (−0.258 + 0.965i)43-s + (0.104 − 0.994i)47-s + (−0.669 − 0.743i)49-s + ⋯
L(s)  = 1  + (−0.0523 + 0.998i)5-s + (−0.406 + 0.913i)7-s + (−0.838 + 0.544i)13-s + (−0.309 + 0.951i)17-s + (−0.987 + 0.156i)19-s + (0.866 + 0.5i)23-s + (−0.994 − 0.104i)25-s + (−0.933 + 0.358i)29-s + (−0.978 − 0.207i)31-s + (−0.891 − 0.453i)35-s + (−0.987 − 0.156i)37-s + (−0.406 − 0.913i)41-s + (−0.258 + 0.965i)43-s + (0.104 − 0.994i)47-s + (−0.669 − 0.743i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3168\)    =    \(2^{5} \cdot 3^{2} \cdot 11\)
Sign: $-0.163 - 0.986i$
Analytic conductor: \(14.7121\)
Root analytic conductor: \(14.7121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3168} (2149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3168,\ (0:\ ),\ -0.163 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07942477404 + 0.09363435866i\)
\(L(\frac12)\) \(\approx\) \(-0.07942477404 + 0.09363435866i\)
\(L(1)\) \(\approx\) \(0.6958242606 + 0.3010139577i\)
\(L(1)\) \(\approx\) \(0.6958242606 + 0.3010139577i\)

Euler product

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

−18.48914322220529049043196793972, −17.49347795601935956919694502274, −17.00671456925890360066240021548, −16.53975027001048083985740116076, −15.75563610585759617859058893825, −15.01178182599802269789169019093, −14.235857133126019658334749158987, −13.2638831440121939166489643949, −13.02537077058800018804888143288, −12.24404189939670974529087932903, −11.43563948309706041853910443603, −10.57597647805757337409055340462, −9.95517484192302618755834248315, −9.11429110344097333269889792114, −8.6112455853696879019332257979, −7.49384296323643588692406350673, −7.16774517076405984021712035711, −6.12682627945404507953418494569, −5.181930834302302255435757457832, −4.62675498887160111118089413046, −3.86447717267925510625775675928, −2.91974893501086271768223138465, −1.94674120479608324513523791808, −0.86437554101752007026445462948, −0.04204140742494954511826979972, 1.971769010651802510580437134, 2.17625534880537844243126454872, 3.36682578145454389751461248504, 3.84922038435420325154354349735, 5.090250600246237797244283815334, 5.73257654830164851084774784432, 6.70695233091062734856114376197, 6.98117774871000977378514049569, 8.065334276490418529209278427612, 8.836671075482058187990955259294, 9.55209849244359883169154408990, 10.26798817209970418224333847438, 11.07772322915194651415002568796, 11.59337539855404536603151589166, 12.59176982688158512858373409050, 12.94791086665865050531525924731, 14.04336523734278021502685714704, 14.77763685285817843939578475588, 15.11990357580041137308744507698, 15.80116812527524225028084599681, 16.85798710906418060873341136105, 17.27962000860591657842883683379, 18.2839860807695612971140703355, 18.75746357832839080748067407939, 19.38700302967191170934014966318

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