Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.973 - 0.227i$
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 + (0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (−0.957 − 0.286i)5-s + (−0.893 − 0.448i)6-s + (−0.686 − 0.727i)7-s + (−0.5 − 0.866i)8-s + (0.973 + 0.230i)9-s + (0.802 + 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.686 + 0.727i)12-s + (−0.686 − 0.727i)13-s + (0.396 + 0.918i)14-s + (−0.918 − 0.396i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s,χ)  = 1  + (−0.939 − 0.342i)2-s + (0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (−0.957 − 0.286i)5-s + (−0.893 − 0.448i)6-s + (−0.686 − 0.727i)7-s + (−0.5 − 0.866i)8-s + (0.973 + 0.230i)9-s + (0.802 + 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.686 + 0.727i)12-s + (−0.686 − 0.727i)13-s + (0.396 + 0.918i)14-s + (−0.918 − 0.396i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.973 - 0.227i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.973 - 0.227i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.973 - 0.227i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1155, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.973 - 0.227i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.05929960653 - 0.5133527269i$
$L(\frac12,\chi)$  $\approx$  $0.05929960653 - 0.5133527269i$
$L(\chi,1)$  $\approx$  0.6456679858 - 0.2037124030i
$L(1,\chi)$  $\approx$  0.6456679858 - 0.2037124030i

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

−18.88888263037963707195528048779, −18.38001554351464340263975915859, −17.58091343176437582685361698827, −16.4374789615753159867752548725, −16.12524114166691039416517640585, −15.31321221815219263493678915710, −14.99172960747568741351634541923, −14.3766199989008483552407300941, −13.23119843705190828204335654333, −12.72838816209156811265695508244, −11.758044270080710379092258257144, −11.1347918043959252868827146889, −10.353073751457323050501739808601, −9.53341611342798790734087226443, −8.960537366836412768464703889021, −8.40004357726405441003488100628, −7.743306597211993464690785216937, −7.020891854435737773363398033085, −6.55778692197112228768328565889, −5.48605079501721972883815878956, −4.54340476269529840714352086889, −3.44040839634096623127493795220, −2.785055707916513810697933202377, −2.259789183025626602878144064865, −1.06244563128141430322250773840, 0.2049744849868935383856843393, 1.102955926381855324678167762490, 2.2856058685145533395468780554, 2.97063142235214484458097701756, 3.53961435316904555099921311119, 4.35048147219936200856209025459, 5.20038565737373568467875796122, 6.74289064745608383623320477295, 7.26742943149937242308328649208, 7.80920140009429602145446675065, 8.2703921617786962832702035483, 9.29202810396633564743628077775, 9.68767933409311342087674252395, 10.4853581392265763269165685305, 10.92566272290476703471156793278, 12.19909123168007132967672682259, 12.51748990303728185607097175995, 13.17600652078289573854618957574, 14.07717493193575244683805096255, 15.03275899005522342280439961348, 15.6066281273327423555535529197, 16.05720719998085315835418787559, 16.72694139845865906823103345013, 17.5362857412556673332225882586, 18.38623387847730589110176681454

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