Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.894 + 0.447i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (0.686 − 0.727i)3-s + 4-s + (0.549 − 0.835i)5-s + (−0.686 + 0.727i)6-s + (−0.0581 + 0.998i)7-s − 8-s + (−0.0581 − 0.998i)9-s + (−0.549 + 0.835i)10-s + (0.549 + 0.835i)11-s + (0.686 − 0.727i)12-s + (0.835 + 0.549i)13-s + (0.0581 − 0.998i)14-s + (−0.230 − 0.973i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (0.686 − 0.727i)3-s + 4-s + (0.549 − 0.835i)5-s + (−0.686 + 0.727i)6-s + (−0.0581 + 0.998i)7-s − 8-s + (−0.0581 − 0.998i)9-s + (−0.549 + 0.835i)10-s + (0.549 + 0.835i)11-s + (0.686 − 0.727i)12-s + (0.835 + 0.549i)13-s + (0.0581 − 0.998i)14-s + (−0.230 − 0.973i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.894 + 0.447i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.894 + 0.447i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.894 + 0.447i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (3039, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ -0.894 + 0.447i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.04998028989 - 0.2113933266i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.04998028989 - 0.2113933266i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7454350112 - 0.2208434227i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7454350112 - 0.2208434227i\)

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

−18.91873335102046337355294362803, −18.15542522736039590638236517614, −17.4988762424314050083421832715, −16.79238208507063935098432619309, −16.329861389670344373557699881497, −15.45867136776712201762660765309, −14.87393846310751791124508187391, −14.30497284691033743802095329473, −13.43239606437545252968394726603, −13.010195608457602292909179107946, −11.41069829912891314856159169431, −10.947326341678168309425646656059, −10.6030405973505250939072136228, −9.84689034116453105114105136607, −9.24188419400397418810173998189, −8.50040292292375503783047795793, −7.86732801869378311005948039986, −7.14014434250314752103769193960, −6.20774281920827134756650700614, −5.85399875219295998573707400017, −4.386837050278211995701649370268, −3.49907119246579010916406852815, −3.18821608019450880876607583539, −2.039393361573903341914693393994, −1.44792628959034451923699674525, 0.0656807288074983575424538379, 1.50458513043540764627631753687, 1.819789802521870031522634351993, 2.42063064834249546766188792476, 3.51855080560686191454512088914, 4.48130974453318351388644908445, 5.7777337722987594811183721995, 6.17034726908841615416451181917, 6.97775061834050577840671921021, 7.74151358216450206227529051151, 8.61667958766395044266558709450, 8.963428712463175639777426461086, 9.40276321786611943404472322906, 10.17109610410285604850358055306, 11.32402822330131630039173092387, 11.98282375144461739062961728661, 12.53945682062388739036346755105, 13.061086333400372391451965507246, 14.18799534485439771750538424035, 14.573489900339945112847466499450, 15.64983401545843780350959836742, 15.95770972464213260703224243930, 16.97767396034768148333745639504, 17.48044938613031061956372710540, 18.36722210941826534674773600895

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