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Properties

Label 1-1699-1699.1062-r1-0-0
Degree $1$
Conductor $1699$
Sign $-0.217 + 0.976i$
Analytic cond. $182.582$
Root an. cond. $182.582$
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  + (0.735 + 0.677i)2-s + (−0.424 − 0.905i)3-s + (0.0831 + 0.996i)4-s + (−0.434 + 0.900i)5-s + (0.300 − 0.953i)6-s + (−0.203 − 0.978i)7-s + (−0.613 + 0.789i)8-s + (−0.639 + 0.768i)9-s + (−0.929 + 0.368i)10-s + (−0.999 + 0.0443i)11-s + (0.866 − 0.498i)12-s + (−0.247 − 0.968i)13-s + (0.512 − 0.858i)14-s + (0.999 + 0.0111i)15-s + (−0.986 + 0.165i)16-s + (0.728 + 0.685i)17-s + ⋯
L(s)  = 1  + (0.735 + 0.677i)2-s + (−0.424 − 0.905i)3-s + (0.0831 + 0.996i)4-s + (−0.434 + 0.900i)5-s + (0.300 − 0.953i)6-s + (−0.203 − 0.978i)7-s + (−0.613 + 0.789i)8-s + (−0.639 + 0.768i)9-s + (−0.929 + 0.368i)10-s + (−0.999 + 0.0443i)11-s + (0.866 − 0.498i)12-s + (−0.247 − 0.968i)13-s + (0.512 − 0.858i)14-s + (0.999 + 0.0111i)15-s + (−0.986 + 0.165i)16-s + (0.728 + 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1699 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1699 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1699\)
Sign: $-0.217 + 0.976i$
Analytic conductor: \(182.582\)
Root analytic conductor: \(182.582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1699} (1062, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1699,\ (1:\ ),\ -0.217 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5769036965 + 0.7194208240i\)
\(L(\frac12)\) \(\approx\) \(0.5769036965 + 0.7194208240i\)
\(L(1)\) \(\approx\) \(0.9345363948 + 0.1802657168i\)
\(L(1)\) \(\approx\) \(0.9345363948 + 0.1802657168i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1699 \( 1 \)
good2 \( 1 + (0.735 + 0.677i)T \)
3 \( 1 + (-0.424 - 0.905i)T \)
5 \( 1 + (-0.434 + 0.900i)T \)
7 \( 1 + (-0.203 - 0.978i)T \)
11 \( 1 + (-0.999 + 0.0443i)T \)
13 \( 1 + (-0.247 - 0.968i)T \)
17 \( 1 + (0.728 + 0.685i)T \)
19 \( 1 + (-0.353 - 0.935i)T \)
23 \( 1 + (-0.999 - 0.0332i)T \)
29 \( 1 + (0.00555 - 0.999i)T \)
31 \( 1 + (0.995 - 0.0997i)T \)
37 \( 1 + (0.937 + 0.347i)T \)
41 \( 1 + (0.997 - 0.0665i)T \)
43 \( 1 + (-0.182 - 0.983i)T \)
47 \( 1 + (-0.728 - 0.685i)T \)
53 \( 1 + (-0.849 - 0.526i)T \)
59 \( 1 + (0.503 + 0.864i)T \)
61 \( 1 + (-0.831 + 0.554i)T \)
67 \( 1 + (0.779 - 0.626i)T \)
71 \( 1 + (-0.225 + 0.974i)T \)
73 \( 1 + (-0.945 - 0.326i)T \)
79 \( 1 + (0.182 - 0.983i)T \)
83 \( 1 + (-0.493 + 0.869i)T \)
89 \( 1 + (0.522 + 0.852i)T \)
97 \( 1 + (0.630 + 0.775i)T \)
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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

−20.22011512918255147238832267790, −19.34341108430115659858248664392, −18.625619146742513559467424422755, −17.86272572480182667725647860313, −16.55134201961524365378110625733, −16.064884815910970192246253767817, −15.64323956282685196289186727792, −14.63622622254271942986465044805, −14.10995389008620315646940743710, −12.80050595098397870910698697924, −12.40904553464658320229316556398, −11.686259657792164562019863844331, −11.14691274455302753481994938537, −9.986292392086486726254175518105, −9.59145726122511696430080725353, −8.758962330882442050462521326217, −7.80020715586659705136081531521, −6.267313764192735319309282501112, −5.70994243420210163897591173006, −4.88421634863035856360507010571, −4.446123314924677097647252784034, −3.42670431543334766809676431526, −2.63000109851261216014958289076, −1.48192748051910345811266100406, −0.21348563556924901201944617769, 0.60347190869480126794490424586, 2.32409272292762678119635612394, 2.95072088145669664178880711224, 3.929051163748323617954830890281, 4.86423279290998615182309499906, 5.87905867587411029774378208536, 6.42487336314835101396134751187, 7.2869063971538802866088466472, 7.847586049148378254886898546551, 8.20790615457219181912184397395, 10.05726722264666622349551731879, 10.6683464362957168660047768748, 11.49566573882174812134055895941, 12.26151418711434015099568658523, 13.11592179986825169061763275565, 13.49657801216098492149565722162, 14.34029520748471881419354785823, 15.11130001148096903688088644620, 15.81994490986747025064101150525, 16.64165505265779543396592772093, 17.52966198986232296088393168475, 17.82684997008337091113111659590, 18.803021648801649536511404450539, 19.58971493381143100506511970216, 20.28671089738850706540339473463

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