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Properties

Label 1-683-683.32-r1-0-0
Degree $1$
Conductor $683$
Sign $0.980 + 0.194i$
Analytic cond. $73.3985$
Root an. cond. $73.3985$
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  + (0.142 + 0.989i)2-s + 3-s + (−0.959 + 0.281i)4-s + (−0.841 − 0.540i)5-s + (0.142 + 0.989i)6-s + (−0.841 − 0.540i)7-s + (−0.415 − 0.909i)8-s + 9-s + (0.415 − 0.909i)10-s + (0.142 + 0.989i)11-s + (−0.959 + 0.281i)12-s + (−0.415 + 0.909i)13-s + (0.415 − 0.909i)14-s + (−0.841 − 0.540i)15-s + (0.841 − 0.540i)16-s + (0.142 + 0.989i)17-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)2-s + 3-s + (−0.959 + 0.281i)4-s + (−0.841 − 0.540i)5-s + (0.142 + 0.989i)6-s + (−0.841 − 0.540i)7-s + (−0.415 − 0.909i)8-s + 9-s + (0.415 − 0.909i)10-s + (0.142 + 0.989i)11-s + (−0.959 + 0.281i)12-s + (−0.415 + 0.909i)13-s + (0.415 − 0.909i)14-s + (−0.841 − 0.540i)15-s + (0.841 − 0.540i)16-s + (0.142 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 683 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 683 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(683\)
Sign: $0.980 + 0.194i$
Analytic conductor: \(73.3985\)
Root analytic conductor: \(73.3985\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{683} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 683,\ (1:\ ),\ 0.980 + 0.194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.602619773 + 0.1573866835i\)
\(L(\frac12)\) \(\approx\) \(1.602619773 + 0.1573866835i\)
\(L(1)\) \(\approx\) \(1.007717153 + 0.3891326183i\)
\(L(1)\) \(\approx\) \(1.007717153 + 0.3891326183i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad683 \( 1 \)
good2 \( 1 + (0.142 + 0.989i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.654 - 0.755i)T \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (-0.654 + 0.755i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−22.34813281275167449862707854876, −21.62050551036718133418942530859, −20.789762925600793008474148232611, −19.777131984943576394097598900537, −19.43711776186460826133673989499, −18.74916397199721083122230957317, −18.146526946196767707722460508752, −16.65322951150141871568822623999, −15.52129070390335512005806290738, −15.031728129584511358905239474236, −14.044890442504895357187660516312, −13.29915495069522719463924430141, −12.46432672504960627159840002889, −11.69494712881886680902104052077, −10.65993003200620850519975400383, −9.889587570018959889961714349689, −8.974392956850141702889507631560, −8.26806870430109376498235236410, −7.2927686216327326292135540841, −6.01508788864570745011245255129, −4.76332042901432213816460218812, −3.48278937610489641688775093593, −3.210586484842560084699407289480, −2.34236858523252127980047728731, −0.789949813214979173402648111933, 0.43377758665351114337679897219, 2.06501621074434259004987573463, 3.66274607693889039521194413294, 4.09519957393488096436753079234, 4.933607253580252099907447559207, 6.70908142610011005356509974352, 6.97104095861978886165345372400, 8.092087720647153726021608099464, 8.706462127700660797249011116494, 9.57587750722870060086807232409, 10.37464929095482026437838327069, 12.21590884371509008401120563441, 12.686109977737338388952264774360, 13.52087761109585684036638238914, 14.417496586215647175410812962975, 15.2696065011763049753269239606, 15.6548241007405536770266948951, 16.851508748520157314975945568921, 17.12105942167279273776019461364, 18.776193942285833633887187804426, 19.16546011781700902147058285913, 19.98631128078794148003858767509, 20.84564937001733203734414419075, 21.81578031461694095453447217973, 22.822773513302571668040180832793

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