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Properties

Label 1-1021-1021.204-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.937 - 0.348i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (0.763 − 0.645i)2-s + (0.739 − 0.673i)3-s + (0.165 − 0.986i)4-s + (0.923 − 0.384i)5-s + (0.128 − 0.991i)6-s + (−0.830 − 0.557i)7-s + (−0.510 − 0.859i)8-s + (0.0922 − 0.995i)9-s + (0.456 − 0.889i)10-s + (0.261 + 0.965i)11-s + (−0.542 − 0.840i)12-s + (0.850 − 0.526i)13-s + (−0.993 + 0.110i)14-s + (0.423 − 0.905i)15-s + (−0.945 − 0.326i)16-s + (−0.823 + 0.567i)17-s + ⋯
L(s)  = 1  + (0.763 − 0.645i)2-s + (0.739 − 0.673i)3-s + (0.165 − 0.986i)4-s + (0.923 − 0.384i)5-s + (0.128 − 0.991i)6-s + (−0.830 − 0.557i)7-s + (−0.510 − 0.859i)8-s + (0.0922 − 0.995i)9-s + (0.456 − 0.889i)10-s + (0.261 + 0.965i)11-s + (−0.542 − 0.840i)12-s + (0.850 − 0.526i)13-s + (−0.993 + 0.110i)14-s + (0.423 − 0.905i)15-s + (−0.945 − 0.326i)16-s + (−0.823 + 0.567i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.937 - 0.348i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.937 - 0.348i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5484238888 - 3.046021089i\)
\(L(\frac12)\) \(\approx\) \(0.5484238888 - 3.046021089i\)
\(L(1)\) \(\approx\) \(1.369414584 - 1.553900089i\)
\(L(1)\) \(\approx\) \(1.369414584 - 1.553900089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.763 - 0.645i)T \)
3 \( 1 + (0.739 - 0.673i)T \)
5 \( 1 + (0.923 - 0.384i)T \)
7 \( 1 + (-0.830 - 0.557i)T \)
11 \( 1 + (0.261 + 0.965i)T \)
13 \( 1 + (0.850 - 0.526i)T \)
17 \( 1 + (-0.823 + 0.567i)T \)
19 \( 1 + (-0.816 + 0.577i)T \)
23 \( 1 + (0.189 - 0.981i)T \)
29 \( 1 + (0.0677 + 0.997i)T \)
31 \( 1 + (0.297 - 0.954i)T \)
37 \( 1 + (-0.949 + 0.314i)T \)
41 \( 1 + (0.816 - 0.577i)T \)
43 \( 1 + (-0.531 - 0.846i)T \)
47 \( 1 + (0.401 - 0.916i)T \)
53 \( 1 + (0.875 + 0.483i)T \)
59 \( 1 + (0.00615 + 0.999i)T \)
61 \( 1 + (0.592 + 0.805i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (0.510 + 0.859i)T \)
73 \( 1 + (-0.993 - 0.110i)T \)
79 \( 1 + (0.631 - 0.775i)T \)
83 \( 1 + (0.213 + 0.976i)T \)
89 \( 1 + (0.552 + 0.833i)T \)
97 \( 1 + (-0.969 + 0.243i)T \)
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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

−21.82883493139186410115273596900, −21.40694873796849255212438068886, −20.83897095704651586274202702375, −19.608056653254965076713396293276, −18.965996614599323894297626075258, −17.882386303202434733827973356683, −17.02801972181640452218845209582, −16.0441625727769008388329525945, −15.75844077632767193976210814395, −14.82447363856371489026493285514, −13.98202978861519431085797663237, −13.49123241256683893143163516130, −12.94445098614444963459497388435, −11.50958212126232053089557297834, −10.85690248548725163302882208536, −9.59314680125979028421896450414, −8.97558721654741224165424214375, −8.37665375019605501580917611761, −6.98465872736749105496814555721, −6.30530357837352333975545584579, −5.58109880726943702440428129071, −4.55412009252631319769595013848, −3.5140742073215038043874591063, −2.89550813628678319674246824574, −2.03181911835269789143800595747, 0.88396563921966670364289545492, 1.86858831695913386206357623209, 2.533063084076688610014891884, 3.68420737755268649684367224179, 4.33851136734403391015022462850, 5.69368306625332075859163954057, 6.476744039767708209799139027086, 7.01453793625406223529385378201, 8.55546503312887690792619248963, 9.180847069591462839867222790843, 10.20839293069612294045348078160, 10.60667447107066455040808689128, 12.19681631560865347912829104932, 12.67533508196574642430430555220, 13.3177694392435209034397704428, 13.78269753723392061955589589310, 14.76211449283505112769042872221, 15.365321725324586963948018373485, 16.55916253746457098127950062414, 17.54259817634706273104337737551, 18.33488331437492998114934215441, 19.12234445232655506801213372699, 19.97299660471994782761609233061, 20.473266375459879163457002996463, 20.96585980522240041495750608387

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