Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.437 - 0.899i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.437 - 0.899i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.437 - 0.899i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.437 - 0.899i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (168, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.437 - 0.899i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9969443733 - 0.6239265095i$
$L(\frac12,\chi)$  $\approx$  $0.9969443733 - 0.6239265095i$
$L(\chi,1)$  $\approx$  0.7204572501 - 0.2562948617i
$L(1,\chi)$  $\approx$  0.7204572501 - 0.2562948617i

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

−18.31202108697763864172739666834, −17.95749424320597392081596613055, −17.065940726484811389388696749080, −16.67808896320499225503807407210, −16.06891515604156125166280047989, −15.49300587886374633161924021372, −14.69559684309560095607574796055, −13.985245472543759198094320408478, −13.02997101429464837515807376385, −12.04545189638870846380701887154, −11.58362719465638586704223968246, −11.020403206552685499732790153795, −9.8530449171021076587609739922, −9.49524521522801011315687615899, −9.283150035121042195322408029622, −8.58944331762995640975083840598, −7.299902357784312821210420229084, −6.598923326060326050531497178888, −5.860043166830686882517432207237, −5.49986689279542789628499102901, −4.31552912002696241703068923530, −3.45229816132032382589072416780, −2.53716705700894796634903678033, −1.7835602377294609538838309523, −0.74987707013160346148598468048, 0.74792961201323650048634972642, 1.23043279951650644559309874422, 2.1744007404236815740404227200, 2.949565266069794910100296579975, 3.7531891075134416072906503378, 5.316642690651263379767817460551, 5.98797220665454897157195698862, 6.52684532935674915947182568412, 7.10326128793014162162173989883, 7.7089496507921233101465817669, 8.84431749394260787821899902311, 9.11420913494248883601721627017, 10.2481599063362012337046403820, 10.67809022199605097274025535875, 11.16649766435392394975963467581, 12.279286934715602489613553311457, 12.75311884812007577403273094189, 13.496511912785171730383839815382, 14.1982563268785115973934888555, 14.90645641610938957404817543350, 16.00944550193293783051140568079, 16.59278445815624162888686513087, 17.25218389638209066312434747111, 17.58848528180341286682079801697, 18.29195832921016601698351491218

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