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Properties

Label 1-1859-1859.1022-r0-0-0
Degree $1$
Conductor $1859$
Sign $-0.157 - 0.987i$
Analytic cond. $8.63315$
Root an. cond. $8.63315$
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  + (−0.992 − 0.120i)2-s + (−0.748 + 0.663i)3-s + (0.970 + 0.239i)4-s + (0.464 + 0.885i)5-s + (0.822 − 0.568i)6-s + (−0.935 − 0.354i)7-s + (−0.935 − 0.354i)8-s + (0.120 − 0.992i)9-s + (−0.354 − 0.935i)10-s + (−0.885 + 0.464i)12-s + (0.885 + 0.464i)14-s + (−0.935 − 0.354i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (−0.239 + 0.970i)18-s − i·19-s + ⋯
L(s)  = 1  + (−0.992 − 0.120i)2-s + (−0.748 + 0.663i)3-s + (0.970 + 0.239i)4-s + (0.464 + 0.885i)5-s + (0.822 − 0.568i)6-s + (−0.935 − 0.354i)7-s + (−0.935 − 0.354i)8-s + (0.120 − 0.992i)9-s + (−0.354 − 0.935i)10-s + (−0.885 + 0.464i)12-s + (0.885 + 0.464i)14-s + (−0.935 − 0.354i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (−0.239 + 0.970i)18-s − i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.157 - 0.987i$
Analytic conductor: \(8.63315\)
Root analytic conductor: \(8.63315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1859,\ (0:\ ),\ -0.157 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04494414063 - 0.05267237529i\)
\(L(\frac12)\) \(\approx\) \(0.04494414063 - 0.05267237529i\)
\(L(1)\) \(\approx\) \(0.4278734924 + 0.1289821746i\)
\(L(1)\) \(\approx\) \(0.4278734924 + 0.1289821746i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.992 - 0.120i)T \)
3 \( 1 + (-0.748 + 0.663i)T \)
5 \( 1 + (0.464 + 0.885i)T \)
7 \( 1 + (-0.935 - 0.354i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 - iT \)
23 \( 1 - T \)
29 \( 1 + (-0.120 + 0.992i)T \)
31 \( 1 + (-0.822 + 0.568i)T \)
37 \( 1 + (0.822 - 0.568i)T \)
41 \( 1 + (0.663 + 0.748i)T \)
43 \( 1 + (0.568 - 0.822i)T \)
47 \( 1 + (0.239 + 0.970i)T \)
53 \( 1 + (-0.354 + 0.935i)T \)
59 \( 1 + (-0.464 - 0.885i)T \)
61 \( 1 + (0.354 + 0.935i)T \)
67 \( 1 + (-0.239 - 0.970i)T \)
71 \( 1 + (-0.663 - 0.748i)T \)
73 \( 1 + (-0.992 + 0.120i)T \)
79 \( 1 + (0.970 - 0.239i)T \)
83 \( 1 + (-0.663 + 0.748i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.464 + 0.885i)T \)
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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

−20.14848917048795898553516757957, −19.41975734005261738258622671181, −18.681099367463045777533824070300, −18.13604036851689810660311872270, −17.42743494175683272287744772146, −16.6735840018383970258964619047, −16.2203573058071641042493350056, −15.693462110618203340579346973238, −14.41787812979410579770439070075, −13.40875545370967278391068862877, −12.781163724854608525999241750900, −11.98677836124686406641323378076, −11.54131136625064300173135621862, −10.374306220900716466649691283997, −9.75779324931411640267308833786, −9.119999101686861720496074033027, −8.16161696721754091084903135445, −7.51002841934730108904717971169, −6.518067103739263222570007875636, −5.92182548828326157386656333586, −5.39675535549693860542549007469, −4.093699932092575206958217736795, −2.59028275212858516508203145733, −1.94673545352835524985166628849, −0.90415859643800059267150922433, 0.04594351395752288307052357718, 1.42088197001791783376248337974, 2.61398702911602067475859539152, 3.384084759726867196673152007166, 4.22652471435081573603529596268, 5.6984753976153214470487018109, 6.24730064669325469181319443354, 6.88047124627455913540297031374, 7.63874391689007167339504166400, 9.054428961990369073375599009367, 9.40149313637244343962647774737, 10.33746671887789313953961247012, 10.70218919262701145102733903874, 11.28857085810418190658895174784, 12.35820959587686470725681408440, 12.98716507496182570963399553357, 14.21225349080596578739446332076, 15.05577131532998840345175689892, 15.7455746672202805375560467607, 16.35286417249953570171656684482, 17.075747647837840958440129686982, 17.79025045866047086355805953050, 18.18876889896937699801988797029, 19.165848969207503230992985782946, 19.81264142094093249636231237704

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