Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.284 - 0.958i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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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 + 5-s + (0.766 − 0.642i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (−0.939 + 0.342i)12-s + (−0.5 + 0.866i)13-s + (−0.939 − 0.342i)14-s + (−0.5 + 0.866i)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 + 5-s + (0.766 − 0.642i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (−0.939 + 0.342i)12-s + (−0.5 + 0.866i)13-s + (−0.939 − 0.342i)14-s + (−0.5 + 0.866i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.284 - 0.958i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.284 - 0.958i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.284 - 0.958i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (3781, \cdot )$
Sato-Tate  :  $\mu(9)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.284 - 0.958i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6502129411 - 0.4853383274i$
$L(\frac12,\chi)$  $\approx$  $0.6502129411 - 0.4853383274i$
$L(\chi,1)$  $\approx$  0.7059622677 + 0.003753982557i
$L(1,\chi)$  $\approx$  0.7059622677 + 0.003753982557i

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

−18.34259121594903297643741150765, −17.850502058147777881933618607676, −17.5802034235922697938724713319, −16.6378026400950717190098729051, −16.37603885114716231114331513703, −15.160715696565899725755479911574, −14.518996886033754084426689095420, −13.98002399745287889745302976327, −13.0620142072541281020448490236, −12.448899697554694150469194803809, −11.573670467596245126175272060308, −10.92495491458784935518019604716, −10.223387618162418330222507487171, −9.84266965382461374995746347893, −8.5792749080990299489555515873, −8.021961985087645551322422039208, −7.657789807949752733369053217552, −6.76303117101493148000569270239, −5.77674812136875348063512956715, −5.57832851096031234522450088126, −4.9383203484367275688073483314, −3.15035389091628776081756457046, −2.33250130865976627673277420022, −1.56338658204932150957825804327, −1.105288897448887321919329616933, 0.33924827630692438556098303588, 1.47276103969237695933459874241, 2.342157471300789656365524807569, 2.88795642477975068659714465073, 4.14137831664541001826199185658, 4.9321943491015150528537256158, 5.45254496333510349952472739388, 6.45840627439853274553846707649, 7.1696395579450000299963894758, 8.00554841812801772678350793982, 8.951054477439850486803724160684, 9.321384425140830045834350518629, 10.204280725333893336199940208336, 10.492463439390791188809208962715, 11.25244699996741677550412823769, 11.97096352822211245634859949952, 12.49952599721963515563102967688, 13.6588166812369013569451255901, 14.32538579250198648526956315073, 15.096990863704381906574840216134, 15.80739889337727424341095980422, 16.56085525818239115060682510003, 17.02099749360087189459840388064, 17.717128481084822600998706415352, 18.15106413083611064478140025584

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