Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.636 - 0.771i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.939 − 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.866 − 0.5i)5-s + (−0.173 − 0.984i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 + 0.984i)12-s + (−0.5 − 0.866i)13-s + (0.766 − 0.642i)14-s i·15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s,χ)  = 1  + (−0.939 − 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.866 − 0.5i)5-s + (−0.173 − 0.984i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 + 0.984i)12-s + (−0.5 − 0.866i)13-s + (0.766 − 0.642i)14-s i·15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.636 - 0.771i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.636 - 0.771i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.636 - 0.771i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2780, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ 0.636 - 0.771i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1099865609 - 0.05186107376i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1099865609 - 0.05186107376i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4644931751 + 0.1483501813i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4644931751 + 0.1483501813i\)

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

−18.5709478452950721641976297799, −18.30822196801670602762390579220, −17.03561867252257107262242052949, −16.70400897064751688698358090136, −15.96733857022492044293520552947, −15.201704707401273708404051428594, −14.51180620212975056116979467220, −13.955341408737136251552271101485, −13.179665144679080883222824070, −12.32829572774965116116179311251, −11.41820958504793854191663746910, −11.13276294743929006613743980940, −10.141956985331964412793121777589, −9.49988945771665604594377286301, −8.56913761763434027630284060784, −8.1050346597650352936311480337, −7.30326138751760385293984004217, −6.94348664956826069867112340750, −6.37634755898442535053471753565, −5.35116878658240380489978219398, −4.09924367389092157678583586910, −3.28833163633276911400676498157, −2.556546052650916154136090189172, −1.72242098185217314409231981472, −0.54747686918357800513225302838, 0.07399794821956828307691747309, 1.66222787067171609662842330248, 2.50620791823933359528561851353, 3.19883219029207783493759450267, 3.78906468684013483654360444022, 4.94568836822495534180166035293, 5.34409841747215026618526496270, 6.71852639748416466334566927117, 7.47578073499186383652115368579, 8.16793353907860106262642472929, 8.71406243207043159794637816881, 9.39547970618921823441251793044, 9.84935414536901410224453761778, 10.77609139685399866285325770504, 11.27658335911582349166033178202, 12.07439872839327121265134952081, 12.98966421020028550714657528299, 13.087383300333629983277171111596, 14.86948262906441022647206773339, 15.23451760247630095006506686099, 15.63173099270780567158938489346, 16.22761207049160281864303754425, 17.00759696362773962968062898431, 17.704832898003042348993898581902, 18.41590180639203185493219187144

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