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Properties

Label 1-684-684.227-r1-0-0
Degree $1$
Conductor $684$
Sign $-0.984 + 0.173i$
Analytic cond. $73.5060$
Root an. cond. $73.5060$
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  + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s − 17-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + 35-s − 37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s − 17-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + 35-s − 37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.984 + 0.173i$
Analytic conductor: \(73.5060\)
Root analytic conductor: \(73.5060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (1:\ ),\ -0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03015647207 - 0.3446900531i\)
\(L(\frac12)\) \(\approx\) \(0.03015647207 - 0.3446900531i\)
\(L(1)\) \(\approx\) \(0.9463750187 - 0.1668714496i\)
\(L(1)\) \(\approx\) \(0.9463750187 - 0.1668714496i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
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

−22.74510264156192907562508999743, −22.32837266868021491371289478426, −21.178730158327047700097450092398, −20.62191963695935547812936651572, −19.76446193978812405477665430827, −18.644898991268050166520627981642, −18.063319236765720962475643690047, −17.33995729562816214572436785127, −16.46060173251359013516047623202, −15.38712370369204295921660252885, −14.602511475260768976913921334103, −13.84472769376842072191609677677, −13.19967160355777460335651724813, −12.01231260981837304377404059738, −10.90749289656764902316499351216, −10.55181691301584543543484154697, −9.5307351829394816096480982327, −8.52308856147648447983364108260, −7.24523631470980213327821294707, −6.902684929168969488176103418191, −5.7377456013006806915801042912, −4.55317674297904320545039942626, −3.76857104607066514584670739711, −2.36781100946832116834162025387, −1.646139525842959580672553675665, 0.07336970215657462972080304698, 1.39472604544022665018068110874, 2.37169423544186887268726326625, 3.56194698092818496525115178297, 4.90258151941596291427435998644, 5.544029283726895558260669030966, 6.27975607996402728071137954943, 7.84059552868831589889549138332, 8.51712066325859188534087900581, 9.16580031791215403889217850208, 10.266937222364612683329751656472, 11.23895793734856118511154387374, 12.0313664026761780034240786416, 13.12843351324467141523043301446, 13.482299626592823226706281369162, 14.66869018447248260008362714737, 15.69386804663975908612165475912, 16.11099862150700873499010281396, 17.35865444222152942451468067473, 17.88674322296951158739294754153, 18.6881212870043263530918056354, 19.77088639553135894730831554327, 20.52020943387097946973156285571, 21.40266567258372280428851214741, 21.766323447318504712791506794161

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