Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.877 + 0.480i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.877 + 0.480i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.877 + 0.480i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.877 + 0.480i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (3067, \cdot )$
Sato-Tate  :  $\mu(27)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.877 + 0.480i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.082221206 + 4.229792652i$
$L(\frac12,\chi)$  $\approx$  $1.082221206 + 4.229792652i$
$L(\chi,1)$  $\approx$  1.778765551 + 1.578221798i
$L(1,\chi)$  $\approx$  1.778765551 + 1.578221798i

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

−18.252300517563031516280170881212, −17.8025960430117041428718097437, −16.898544124631717901437653197710, −15.98223347111946977344360847721, −14.81923240487811391635431687253, −14.60497508066063445514820930664, −14.093282012177862071971940396438, −13.57171354931610891234893626665, −12.89845119299653440400829169589, −11.97186191501589527283395128915, −11.51655188209107517685737135172, −10.50000462269482822850333842657, −9.98632078489462757957595688909, −9.43397661983606798660847345011, −8.50103844248974936503597670273, −7.449053215631724588743166616613, −6.76345832734085042099190328316, −6.50141046394492199056551757276, −5.2366645828101963596774791130, −4.5102520985040630855901958991, −3.58594152773310725442782748218, −3.14570458240265878516091526447, −2.19544127365593728816767669287, −1.692354805839430503063159311895, −0.70912551852945151876134560164, 1.73334846228866960675131362641, 2.03125436228196066719188705982, 3.08557878159317137836269693707, 3.92191037404475187147443376294, 4.571699605066664248979216444291, 5.26784490969243219697618998906, 5.88307141797672000465030597136, 6.70897619963706942741265138704, 7.83095400726970779322513004837, 8.248718710110882456136275841959, 8.849558933137738662288136716234, 9.66577699059913138749588012352, 10.139218442250274509389035946935, 11.52961431984323863179772337643, 12.252918779049513253279185874947, 12.58748441104018316634994378821, 13.46349298271856387235979267973, 14.212222510238288882649619854809, 14.69218611061645352137047964064, 15.06682864136198495925549977699, 15.98935852058676826172408468068, 16.59256259780831287308355162002, 17.15979455563725308570519877215, 17.83394280892469710980194764940, 18.79324868726521146998907250482

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