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Properties

Label 1-517-517.68-r1-0-0
Degree $1$
Conductor $517$
Sign $0.879 - 0.476i$
Analytic cond. $55.5593$
Root an. cond. $55.5593$
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  + (−0.946 + 0.321i)2-s + (−0.839 + 0.542i)3-s + (0.792 − 0.609i)4-s + (−0.981 − 0.190i)5-s + (0.620 − 0.784i)6-s + (0.702 − 0.711i)7-s + (−0.554 + 0.832i)8-s + (0.410 − 0.911i)9-s + (0.990 − 0.136i)10-s + (−0.334 + 0.942i)12-s + (−0.976 − 0.216i)13-s + (−0.435 + 0.900i)14-s + (0.927 − 0.373i)15-s + (0.256 − 0.966i)16-s + (0.996 − 0.0818i)17-s + (−0.0954 + 0.995i)18-s + ⋯
L(s)  = 1  + (−0.946 + 0.321i)2-s + (−0.839 + 0.542i)3-s + (0.792 − 0.609i)4-s + (−0.981 − 0.190i)5-s + (0.620 − 0.784i)6-s + (0.702 − 0.711i)7-s + (−0.554 + 0.832i)8-s + (0.410 − 0.911i)9-s + (0.990 − 0.136i)10-s + (−0.334 + 0.942i)12-s + (−0.976 − 0.216i)13-s + (−0.435 + 0.900i)14-s + (0.927 − 0.373i)15-s + (0.256 − 0.966i)16-s + (0.996 − 0.0818i)17-s + (−0.0954 + 0.995i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.879 - 0.476i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.879 - 0.476i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5987634802 - 0.1518292574i\)
\(L(\frac12)\) \(\approx\) \(0.5987634802 - 0.1518292574i\)
\(L(1)\) \(\approx\) \(0.4926797253 + 0.05028356688i\)
\(L(1)\) \(\approx\) \(0.4926797253 + 0.05028356688i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.946 + 0.321i)T \)
3 \( 1 + (-0.839 + 0.542i)T \)
5 \( 1 + (-0.981 - 0.190i)T \)
7 \( 1 + (0.702 - 0.711i)T \)
13 \( 1 + (-0.976 - 0.216i)T \)
17 \( 1 + (0.996 - 0.0818i)T \)
19 \( 1 + (0.484 + 0.874i)T \)
23 \( 1 + (-0.576 - 0.816i)T \)
29 \( 1 + (-0.0954 + 0.995i)T \)
31 \( 1 + (0.998 - 0.0546i)T \)
37 \( 1 + (-0.435 - 0.900i)T \)
41 \( 1 + (-0.0409 + 0.999i)T \)
43 \( 1 + (0.334 + 0.942i)T \)
53 \( 1 + (-0.937 - 0.347i)T \)
59 \( 1 + (-0.999 - 0.0273i)T \)
61 \( 1 + (-0.881 + 0.472i)T \)
67 \( 1 + (0.460 - 0.887i)T \)
71 \( 1 + (0.946 + 0.321i)T \)
73 \( 1 + (-0.994 + 0.109i)T \)
79 \( 1 + (0.531 - 0.847i)T \)
83 \( 1 + (0.385 - 0.922i)T \)
89 \( 1 + (0.682 + 0.730i)T \)
97 \( 1 + (0.256 + 0.966i)T \)
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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

−23.65763642721593498001400262972, −22.42222911413967137845509243100, −21.78656360431988684920011133981, −20.80221297921680882370477200074, −19.67138275744829070532001294732, −19.02780515236377342185229298931, −18.478880786708231801507795072928, −17.46894976644426039387643840789, −16.99645452093403373719549054041, −15.775898374234944652084094451756, −15.327627017952835150989636211951, −13.92485343873765894706104423530, −12.412266848114937598902023122431, −11.94548758938855013245598025288, −11.44892948288644214410775918699, −10.48797543347567063702956669334, −9.454193042830947232977210835252, −8.16426991201203132815502223416, −7.65054536764339617734890798361, −6.82420999582950137717910083542, −5.59540039083332703441181295062, −4.47988015045332001397650363528, −2.98211248163576085642408466906, −1.891359748354105225005392590791, −0.688748153534143301971378410192, 0.41490726061067625099620389809, 1.35890116632912536112428891055, 3.245179702620519419950414974412, 4.51512401074166038620770192123, 5.28071799333808168738128157673, 6.49850698506502343894130433903, 7.57387554498118700512850701554, 8.034267223024563454137165438916, 9.37487227359040237678315239327, 10.28074458243637973544913572898, 10.882171689332837927174364795413, 11.88133824179151801969281186655, 12.356348055059194422427685392998, 14.39739817621339207454362949562, 14.84531519201768443552011554355, 16.02573391372095922196823441487, 16.47980976781291757130697582181, 17.21668532083759156699816655925, 18.05461054984028333505069903717, 18.89694059465238067830607253010, 19.96943554596153041574995253950, 20.54214699550966131699129557452, 21.41914338195328764968205339316, 22.81593424222275793061009587351, 23.27192668713129127257625076745

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