Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.0326 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (−0.396 + 0.918i)3-s + 4-s + (−0.957 + 0.286i)5-s + (0.396 − 0.918i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (0.957 − 0.286i)10-s + (−0.957 − 0.286i)11-s + (−0.396 + 0.918i)12-s + (0.286 + 0.957i)13-s + (0.686 − 0.727i)14-s + (0.116 − 0.993i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (−0.396 + 0.918i)3-s + 4-s + (−0.957 + 0.286i)5-s + (0.396 − 0.918i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (0.957 − 0.286i)10-s + (−0.957 − 0.286i)11-s + (−0.396 + 0.918i)12-s + (0.286 + 0.957i)13-s + (0.686 − 0.727i)14-s + (0.116 − 0.993i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0326 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0326 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.0326 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (24, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.0326 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1243635628 + 0.1284907814i$
$L(\frac12,\chi)$  $\approx$  $-0.1243635628 + 0.1284907814i$
$L(\chi,1)$  $\approx$  0.3362706049 + 0.2192599153i
$L(1,\chi)$  $\approx$  0.3362706049 + 0.2192599153i

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

−17.89170573997179846342738251674, −17.50132818055455343688247234448, −16.58639219013946885172233360195, −16.1974752669330369639417156085, −15.461737557169137226179241017849, −14.86381783443558403771752365340, −13.55873656193185691866900025701, −12.95914313973377136401427396945, −12.50817893601779414540070159698, −11.63876291483353613382078065536, −11.07225333239434621633638340531, −10.37133318251465147473027390150, −9.82110304036272708442265004540, −8.54925432758103738308029133678, −8.110750406212808674822973346919, −7.63707772980790839173563287441, −6.843551299720423214420068864294, −6.341772051723443977092319824876, −5.38815091496642917664621880140, −4.45072757697225147889943599296, −3.22178337198068639150925829975, −2.749851126512220705624973526599, −1.58782526444239784444723288487, −0.64173392877324068024552203705, −0.13326033663394348675450062582, 1.12297723603353332018706479637, 2.55453496232978239372206293580, 3.13090971945304063001429768184, 3.73491894415442639266014880583, 4.97846758020171758364886748552, 5.51976562454552221880215225504, 6.61293631156581745921545049534, 6.97077523365755519305625372578, 8.0466415678153582134623511196, 8.69394837761773369128582829028, 9.29011610059175085188455840047, 9.97731248680740227412775052934, 10.633199522090498811410057576787, 11.40499018246386773130018726682, 11.79071588058903084876769173436, 12.361221396234228513649887192143, 13.59083061390302821811331902056, 14.52551875401638966388648569624, 15.379355544461449199768459122450, 15.90656354904503796000866462589, 16.02730414355467531163615574313, 16.69255597357994912012310238301, 17.824989022848339032694584426303, 18.21513953599116885423273933602, 18.98254599307310691957595693959

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