Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.0519 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0519 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0519 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.0519 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (695, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.0519 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.120536701 + 1.063763227i$
$L(\frac12,\chi)$  $\approx$  $1.120536701 + 1.063763227i$
$L(\chi,1)$  $\approx$  0.9757752676 + 0.3288048127i
$L(1,\chi)$  $\approx$  0.9757752676 + 0.3288048127i

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

−18.424640012354079093740437773550, −17.32329312668811270545003365610, −16.959226794171855043054059597457, −16.68770501394476713634124525855, −15.76470567266725228719824299952, −14.73695991942886930529493031659, −14.000076170389351036458604392225, −13.59937293058575846128606062639, −12.76379386389441980991148413010, −12.21051207436108495006385035202, −11.15773333761967149425986079136, −10.40831830487542990891894090423, −9.9943885526491243052812501060, −9.59194438183971521067098549214, −8.714505134763351700138862555833, −8.16246138450870994365397563372, −7.44207211419333985802177380169, −6.11852824469518238651567665316, −5.55134094315818785083126500438, −4.28562322867346483804471633387, −4.017629162804988625015991962352, −3.20842090843968420280697657685, −2.34491115229370398355802423652, −1.53912905465876797668198865067, −0.53994759311480281834018374295, 0.960017632914202236735408489260, 2.02020647480480776556236039668, 2.33837140259843939844107723352, 3.46311874481076447487336188244, 4.633597883653233681441011991, 5.54283570173634672386884310003, 6.27855484077349518889623152298, 6.74200579622712637997784096762, 7.22742704093980080509867481868, 8.14839870784110131042074961688, 9.085584508029104012072919108314, 9.38570979244443517947912373672, 9.840930654867191685807316604, 11.04372793519518122608712895130, 11.895104969613552719954554692409, 12.60180825640118598499237993138, 13.3883467190405836434482340775, 14.16789103940642282032868007268, 14.41337294663846291448772770239, 15.11046096580317477128119446132, 15.90421053339819557145807614481, 16.76245286575468155443747937578, 17.40898402030180433077279397180, 18.01397716874103697882715481019, 18.45168728308403987451437267799

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