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Properties

Label 1-1805-1805.12-r0-0-0
Degree $1$
Conductor $1805$
Sign $0.887 + 0.461i$
Analytic cond. $8.38238$
Root an. cond. $8.38238$
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  + (−0.523 − 0.851i)2-s + (0.936 + 0.350i)3-s + (−0.451 + 0.892i)4-s + (−0.191 − 0.981i)6-s + (0.837 + 0.546i)7-s + (0.996 − 0.0825i)8-s + (0.754 + 0.656i)9-s + (0.945 + 0.324i)11-s + (−0.735 + 0.677i)12-s + (−0.771 + 0.635i)13-s + (0.0275 − 0.999i)14-s + (−0.592 − 0.805i)16-s + (−0.892 + 0.451i)17-s + (0.164 − 0.986i)18-s + (0.592 + 0.805i)21-s + (−0.218 − 0.975i)22-s + ⋯
L(s)  = 1  + (−0.523 − 0.851i)2-s + (0.936 + 0.350i)3-s + (−0.451 + 0.892i)4-s + (−0.191 − 0.981i)6-s + (0.837 + 0.546i)7-s + (0.996 − 0.0825i)8-s + (0.754 + 0.656i)9-s + (0.945 + 0.324i)11-s + (−0.735 + 0.677i)12-s + (−0.771 + 0.635i)13-s + (0.0275 − 0.999i)14-s + (−0.592 − 0.805i)16-s + (−0.892 + 0.451i)17-s + (0.164 − 0.986i)18-s + (0.592 + 0.805i)21-s + (−0.218 − 0.975i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.887 + 0.461i$
Analytic conductor: \(8.38238\)
Root analytic conductor: \(8.38238\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1805,\ (0:\ ),\ 0.887 + 0.461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.884981347 + 0.4608902031i\)
\(L(\frac12)\) \(\approx\) \(1.884981347 + 0.4608902031i\)
\(L(1)\) \(\approx\) \(1.255951273 + 0.02288785218i\)
\(L(1)\) \(\approx\) \(1.255951273 + 0.02288785218i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.523 - 0.851i)T \)
3 \( 1 + (0.936 + 0.350i)T \)
7 \( 1 + (0.837 + 0.546i)T \)
11 \( 1 + (0.945 + 0.324i)T \)
13 \( 1 + (-0.771 + 0.635i)T \)
17 \( 1 + (-0.892 + 0.451i)T \)
23 \( 1 + (0.936 - 0.350i)T \)
29 \( 1 + (-0.998 - 0.0550i)T \)
31 \( 1 + (0.879 - 0.475i)T \)
37 \( 1 + (0.324 - 0.945i)T \)
41 \( 1 + (-0.716 - 0.697i)T \)
43 \( 1 + (0.376 + 0.926i)T \)
47 \( 1 + (0.981 - 0.191i)T \)
53 \( 1 + (0.981 - 0.191i)T \)
59 \( 1 + (0.716 + 0.697i)T \)
61 \( 1 + (0.904 - 0.426i)T \)
67 \( 1 + (0.569 + 0.821i)T \)
71 \( 1 + (-0.904 - 0.426i)T \)
73 \( 1 + (0.892 - 0.451i)T \)
79 \( 1 + (-0.926 + 0.376i)T \)
83 \( 1 + (-0.915 + 0.401i)T \)
89 \( 1 + (0.451 - 0.892i)T \)
97 \( 1 + (-0.569 + 0.821i)T \)
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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

−20.1865779342580966257604737292, −19.25883157431266037473866069448, −18.71972971223154033470670300814, −17.74956636380394190479701158676, −17.32927326651637621491731746450, −16.5938152230152474792155847618, −15.43061770570434839670104762750, −15.02277811545343503017346424351, −14.32389865617733150660343944222, −13.68899510810433106833226175645, −13.090653723241726580552474030860, −11.866738128542783348634255783691, −10.99964619162268879589853030769, −10.06265447726423036973975431035, −9.33683608150589226345863538117, −8.59627965191251815513052689513, −8.038483869911878030437178114802, −7.056760652505654472782251320, −6.874244895445090663030232675472, −5.5477505955206339513177776908, −4.651682382109348423366258211033, −3.88754606199650254525598266367, −2.654303450067374644952043390403, −1.58260728673942254163447753227, −0.80490271785414526071332311479, 1.26457453286858671393724821287, 2.17467207506981290839275595000, 2.56258988593519804058165292451, 3.93749171522486445504699700076, 4.29977934293031712443112582750, 5.23511116908452872747943055226, 6.82412094274024279955501152830, 7.52314099207102723806285577672, 8.4491067936268665082412153511, 9.01231298592946863660897928940, 9.490668739588225257316645139019, 10.42563369519220141421205780678, 11.24564752897305447257711533637, 11.88300259470321439823244738302, 12.74204084147117611748132782483, 13.49329436674505374317669025956, 14.44628663289522171844801485395, 14.84153573621155827692342244361, 15.76863820720939071710145329994, 16.86432545680001025805535306250, 17.31981027089425675052755907758, 18.243357077496800460125414624188, 19.03626191155864332763108320158, 19.4826115039555020490205952268, 20.24416441011384979215312668281

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