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Properties

Label 1-2833-2833.1028-r1-0-0
Degree $1$
Conductor $2833$
Sign $-0.620 - 0.784i$
Analytic cond. $304.448$
Root an. cond. $304.448$
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  + (−0.388 + 0.921i)2-s + (−0.891 + 0.453i)3-s + (−0.697 − 0.716i)4-s + (0.615 − 0.787i)5-s + (−0.0709 − 0.997i)6-s + (0.0465 − 0.998i)7-s + (0.931 − 0.364i)8-s + (0.589 − 0.807i)9-s + (0.486 + 0.873i)10-s + (0.295 − 0.955i)11-s + (0.946 + 0.322i)12-s + (0.999 − 0.0310i)13-s + (0.902 + 0.431i)14-s + (−0.191 + 0.981i)15-s + (−0.0266 + 0.999i)16-s + (0.345 − 0.938i)17-s + ⋯
L(s)  = 1  + (−0.388 + 0.921i)2-s + (−0.891 + 0.453i)3-s + (−0.697 − 0.716i)4-s + (0.615 − 0.787i)5-s + (−0.0709 − 0.997i)6-s + (0.0465 − 0.998i)7-s + (0.931 − 0.364i)8-s + (0.589 − 0.807i)9-s + (0.486 + 0.873i)10-s + (0.295 − 0.955i)11-s + (0.946 + 0.322i)12-s + (0.999 − 0.0310i)13-s + (0.902 + 0.431i)14-s + (−0.191 + 0.981i)15-s + (−0.0266 + 0.999i)16-s + (0.345 − 0.938i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2833 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2833 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2833\)
Sign: $-0.620 - 0.784i$
Analytic conductor: \(304.448\)
Root analytic conductor: \(304.448\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2833} (1028, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2833,\ (1:\ ),\ -0.620 - 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5133540165 - 1.061124960i\)
\(L(\frac12)\) \(\approx\) \(0.5133540165 - 1.061124960i\)
\(L(1)\) \(\approx\) \(0.7600769415 - 0.03229501569i\)
\(L(1)\) \(\approx\) \(0.7600769415 - 0.03229501569i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2833 \( 1 \)
good2 \( 1 + (-0.388 + 0.921i)T \)
3 \( 1 + (-0.891 + 0.453i)T \)
5 \( 1 + (0.615 - 0.787i)T \)
7 \( 1 + (0.0465 - 0.998i)T \)
11 \( 1 + (0.295 - 0.955i)T \)
13 \( 1 + (0.999 - 0.0310i)T \)
17 \( 1 + (0.345 - 0.938i)T \)
19 \( 1 + (-0.619 + 0.785i)T \)
23 \( 1 + (0.903 - 0.429i)T \)
29 \( 1 + (0.990 + 0.137i)T \)
31 \( 1 + (0.307 - 0.951i)T \)
37 \( 1 + (-0.314 - 0.949i)T \)
41 \( 1 + (0.141 - 0.989i)T \)
43 \( 1 + (-0.421 + 0.906i)T \)
47 \( 1 + (-0.853 - 0.520i)T \)
53 \( 1 + (-0.912 + 0.409i)T \)
59 \( 1 + (0.429 - 0.903i)T \)
61 \( 1 + (-0.979 - 0.202i)T \)
67 \( 1 + (-0.638 + 0.769i)T \)
71 \( 1 + (0.970 - 0.241i)T \)
73 \( 1 + (-0.583 - 0.811i)T \)
79 \( 1 + (-0.427 + 0.904i)T \)
83 \( 1 + (0.437 + 0.899i)T \)
89 \( 1 + (-0.230 - 0.972i)T \)
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

−19.14906083290466976387059093734, −18.50735887839634261477642539008, −17.90620479839473425351320396194, −17.49214136326624057350125101888, −16.8930990896642984050876540405, −15.775625489593720436031054472213, −15.061586296716920066338158148570, −14.14762797351118489459820439815, −13.17558342534830040276314512521, −12.87475546379654499202056497543, −11.93011493157685032282203055089, −11.506757859260132302708652818790, −10.62787236508891148169995615951, −10.26281368492881591743756726069, −9.329548648622130034176267728796, −8.59709152630189991879808628715, −7.73559159730393816046988483407, −6.66481099073018748062003606925, −6.30649926418589203436910686664, −5.20607023137005024139111574599, −4.59010026944523576108569888128, −3.3660308242775052719241970922, −2.585752059664797371770383181768, −1.68667756948010853171191668354, −1.26231860546739004005934934476, 0.321147159350604381744041760652, 0.837629412049855107327111906960, 1.53629127302928003431137108116, 3.417285943266561755689397545, 4.2762375510162890702839402103, 4.87245177165378157388760393924, 5.703050011167498446855514709670, 6.25327415336974102068957699773, 6.87411151784449204117653640002, 7.94938827311693154463460682435, 8.67394321688778955153469583949, 9.37939022431317278048314264768, 10.0901393239155199541606985252, 10.73525322148298689601997069611, 11.36900045142744850673656574131, 12.52936372353944177950466948987, 13.248441406875142691477523516145, 13.91987855223017833315207709303, 14.485615210310043596010942504243, 15.623327166130096949818352659511, 16.27185056642340565384149904369, 16.60575262971697807003924687117, 17.14436147455754281614658942249, 17.79218363291448568119493087182, 18.4984140180661429907574217614

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