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
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: | $\chi_{683} (619, \cdot )$ |
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
$p$ | $F_p(T)$ | |
---|---|---|
bad | 683 | \( 1 \) |
good | 2 | \( 1 + (0.142 - 0.989i)T \) |
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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Imaginary part of the first few zeros on the critical line
−22.822773513302571668040180832793, −21.81578031461694095453447217973, −20.84564937001733203734414419075, −19.98631128078794148003858767509, −19.16546011781700902147058285913, −18.776193942285833633887187804426, −17.12105942167279273776019461364, −16.851508748520157314975945568921, −15.6548241007405536770266948951, −15.2696065011763049753269239606, −14.417496586215647175410812962975, −13.52087761109585684036638238914, −12.686109977737338388952264774360, −12.21590884371509008401120563441, −10.37464929095482026437838327069, −9.57587750722870060086807232409, −8.706462127700660797249011116494, −8.092087720647153726021608099464, −6.97104095861978886165345372400, −6.70908142610011005356509974352, −4.933607253580252099907447559207, −4.09519957393488096436753079234, −3.66274607693889039521194413294, −2.06501621074434259004987573463, −0.43377758665351114337679897219, 0.789949813214979173402648111933, 2.34236858523252127980047728731, 3.210586484842560084699407289480, 3.48278937610489641688775093593, 4.76332042901432213816460218812, 6.01508788864570745011245255129, 7.2927686216327326292135540841, 8.26806870430109376498235236410, 8.974392956850141702889507631560, 9.889587570018959889961714349689, 10.65993003200620850519975400383, 11.69494712881886680902104052077, 12.46432672504960627159840002889, 13.29915495069522719463924430141, 14.044890442504895357187660516312, 15.031728129584511358905239474236, 15.52129070390335512005806290738, 16.65322951150141871568822623999, 18.146526946196767707722460508752, 18.74916397199721083122230957317, 19.43711776186460826133673989499, 19.777131984943576394097598900537, 20.789762925600793008474148232611, 21.62050551036718133418942530859, 22.34813281275167449862707854876