Properties

Degree 1
Conductor $ 7 \cdot 571 $
Sign $0.997 + 0.0700i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.350 + 0.936i)2-s + (0.997 + 0.0770i)3-s + (−0.754 + 0.656i)4-s + (−0.411 − 0.911i)5-s + (0.277 + 0.960i)6-s + (−0.879 − 0.475i)8-s + (0.988 + 0.153i)9-s + (0.709 − 0.705i)10-s + (−0.256 + 0.966i)11-s + (−0.802 + 0.596i)12-s + (0.441 − 0.897i)13-s + (−0.340 − 0.940i)15-s + (0.137 − 0.990i)16-s + (0.899 + 0.436i)17-s + (0.202 + 0.979i)18-s + (−0.761 − 0.648i)19-s + ⋯
L(s,χ)  = 1  + (0.350 + 0.936i)2-s + (0.997 + 0.0770i)3-s + (−0.754 + 0.656i)4-s + (−0.411 − 0.911i)5-s + (0.277 + 0.960i)6-s + (−0.879 − 0.475i)8-s + (0.988 + 0.153i)9-s + (0.709 − 0.705i)10-s + (−0.256 + 0.966i)11-s + (−0.802 + 0.596i)12-s + (0.441 − 0.897i)13-s + (−0.340 − 0.940i)15-s + (0.137 − 0.990i)16-s + (0.899 + 0.436i)17-s + (0.202 + 0.979i)18-s + (−0.761 − 0.648i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.997 + 0.0700i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.997 + 0.0700i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(3997\)    =    \(7 \cdot 571\)
\( \varepsilon \)  =  $0.997 + 0.0700i$
motivic weight  =  \(0\)
character  :  $\chi_{3997} (97, \cdot )$
Sato-Tate  :  $\mu(570)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 3997,\ (1:\ ),\ 0.997 + 0.0700i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(3.114603430 + 0.1092000238i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(3.114603430 + 0.1092000238i\)
\(L(\chi,1)\)  \(\approx\)  \(1.426189661 + 0.5114195451i\)
\(L(1,\chi)\)  \(\approx\)  \(1.426189661 + 0.5114195451i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.70147285117862932976333206874, −18.21236802588208545216712476380, −16.91590395168163348985817048111, −16.12923079912381266301943026745, −15.26255629641227911885938322499, −14.636853998415597372702947049992, −14.18422567066450619453500390042, −13.630382267624424165152300098802, −12.86508613640706372645119219306, −12.196921419830196708872534499427, −11.177116869455319113774342877179, −10.98122277998231778214089113129, −10.01683344665146750869588826877, −9.42107770715433232900771069470, −8.6078525416708958798852893693, −8.04452144774495050639086023906, −7.10984174016158471084474762671, −6.34280159567882986423634641696, −5.458719112241944529218978905434, −4.40408323601246787655251319488, −3.602393432073312537989520112146, −3.320178661774372901276305226443, −2.470240346038034948906237852655, −1.73912575576576288216613788040, −0.77918291579023819788027482947, 0.42356780048249490311933427220, 1.459224457420255330677316785255, 2.546151536466780569866817294783, 3.589247275887504474991668526448, 3.938596857342355001175112469324, 5.00141454638351971269730220815, 5.26394348350248249924425403968, 6.47318808914128759525363438144, 7.33881017056090309742991130797, 7.83485900730613177428143290198, 8.41892696556930739268068283687, 9.075502715494743068599749087303, 9.67712666828609445601072478296, 10.53022921109094031482602380818, 11.71284949919224506291044959775, 12.64354813041648207393673961634, 13.058395718964335193776635114290, 13.32253802310621514851852829232, 14.53150964207592678200355900335, 15.04980470323793672494288425914, 15.3818233401545616811499766, 16.10314228464383521118040883974, 16.960349785358981457728712045061, 17.30351867925048478803615917088, 18.47763718157715431474305600576

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