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Properties

Label 1-1859-1859.354-r1-0-0
Degree $1$
Conductor $1859$
Sign $0.861 + 0.507i$
Analytic cond. $199.777$
Root an. cond. $199.777$
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  + (0.704 + 0.709i)2-s + (0.991 − 0.128i)3-s + (−0.00805 + 0.999i)4-s + (−0.681 − 0.732i)5-s + (0.789 + 0.613i)6-s + (−0.937 + 0.347i)7-s + (−0.715 + 0.698i)8-s + (0.966 − 0.254i)9-s + (0.0402 − 0.999i)10-s + (0.120 + 0.992i)12-s + (−0.906 − 0.421i)14-s + (−0.769 − 0.638i)15-s + (−0.999 − 0.0161i)16-s + (−0.619 − 0.784i)17-s + (0.861 + 0.506i)18-s + (0.978 − 0.207i)19-s + ⋯
L(s)  = 1  + (0.704 + 0.709i)2-s + (0.991 − 0.128i)3-s + (−0.00805 + 0.999i)4-s + (−0.681 − 0.732i)5-s + (0.789 + 0.613i)6-s + (−0.937 + 0.347i)7-s + (−0.715 + 0.698i)8-s + (0.966 − 0.254i)9-s + (0.0402 − 0.999i)10-s + (0.120 + 0.992i)12-s + (−0.906 − 0.421i)14-s + (−0.769 − 0.638i)15-s + (−0.999 − 0.0161i)16-s + (−0.619 − 0.784i)17-s + (0.861 + 0.506i)18-s + (0.978 − 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.861 + 0.507i$
Analytic conductor: \(199.777\)
Root analytic conductor: \(199.777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (354, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1859,\ (1:\ ),\ 0.861 + 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.207920375 + 0.8750444583i\)
\(L(\frac12)\) \(\approx\) \(3.207920375 + 0.8750444583i\)
\(L(1)\) \(\approx\) \(1.588620824 + 0.5005309576i\)
\(L(1)\) \(\approx\) \(1.588620824 + 0.5005309576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.704 + 0.709i)T \)
3 \( 1 + (0.991 - 0.128i)T \)
5 \( 1 + (-0.681 - 0.732i)T \)
7 \( 1 + (-0.937 + 0.347i)T \)
17 \( 1 + (-0.619 - 0.784i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.152 - 0.988i)T \)
31 \( 1 + (0.644 + 0.764i)T \)
37 \( 1 + (0.899 + 0.435i)T \)
41 \( 1 + (-0.991 + 0.128i)T \)
43 \( 1 + (-0.692 - 0.721i)T \)
47 \( 1 + (-0.861 + 0.506i)T \)
53 \( 1 + (-0.443 - 0.896i)T \)
59 \( 1 + (0.818 + 0.574i)T \)
61 \( 1 + (-0.997 + 0.0643i)T \)
67 \( 1 + (0.948 - 0.316i)T \)
71 \( 1 + (0.853 - 0.520i)T \)
73 \( 1 + (0.998 + 0.0483i)T \)
79 \( 1 + (-0.215 - 0.976i)T \)
83 \( 1 + (0.943 + 0.331i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.974 + 0.223i)T \)
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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

−19.96890565694120947619484531850, −19.46917052245603467550447034153, −18.5629965236197998318055150043, −18.301763978703512525296087494069, −16.67148253379037681032674341709, −15.78478724121462132712425648048, −15.36842502937830205670827153897, −14.50476505046287603601873518655, −14.04842982088100263346378536262, −13.14686650912229795650228584198, −12.66154474864594971118197587797, −11.71980745581966810590997339544, −10.85327936689930864082240846206, −10.14558714877368885196976863361, −9.63531219794059600777682228157, −8.618655385511453386536733484166, −7.7056071112946764056935303114, −6.7653548106281875846860331111, −6.23948413938221076119110808918, −4.84080189635588141764018419261, −3.98111423613881874724879058740, −3.4667641589006776984869191942, −2.80644814737698862021296265626, −1.95506571140355835287903008192, −0.68695851978298662176168879766, 0.55466669011446404102599260297, 2.04077980398368354978565057695, 3.12748401236916875441266553519, 3.53594496571329461853673596875, 4.52927230132716704921880488413, 5.2178704754561344301319240649, 6.363322830180927779622512269854, 7.08434298248589017000853978385, 7.84953538097598632473080930493, 8.46944382344534228712900587876, 9.29644200291639661674421322865, 9.79917085369733817812213450057, 11.51095532458039209969330447683, 12.01723938268004533635275417733, 12.833494411708059183144626302035, 13.48982069821261201140078390366, 13.873150148809693723891125402911, 15.08610785307712220848333562675, 15.54517724787027460370862578471, 16.01908228463344439484244202714, 16.70101844049299989004972776677, 17.800646765267585849008021970202, 18.55133681738056399982109464006, 19.52415676512681145397009078437, 20.020148564346163014043213448561

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