Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.283 + 0.958i$
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 + (−0.835 + 0.549i)3-s + (0.766 + 0.642i)4-s + (0.993 + 0.116i)5-s + (−0.973 + 0.230i)6-s + (0.597 − 0.802i)7-s + (0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (−0.993 − 0.116i)12-s + (−0.597 + 0.802i)13-s + (0.835 − 0.549i)14-s + (−0.893 + 0.448i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s,χ)  = 1  + (0.939 + 0.342i)2-s + (−0.835 + 0.549i)3-s + (0.766 + 0.642i)4-s + (0.993 + 0.116i)5-s + (−0.973 + 0.230i)6-s + (0.597 − 0.802i)7-s + (0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (−0.993 − 0.116i)12-s + (−0.597 + 0.802i)13-s + (0.835 − 0.549i)14-s + (−0.893 + 0.448i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.283 + 0.958i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.283 + 0.958i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.283 + 0.958i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (3061, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.283 + 0.958i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.985335505 + 2.230410823i$
$L(\frac12,\chi)$  $\approx$  $2.985335505 + 2.230410823i$
$L(\chi,1)$  $\approx$  1.844220358 + 0.7732023360i
$L(1,\chi)$  $\approx$  1.844220358 + 0.7732023360i

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

−18.28204194758502589879363766458, −17.606236904647358853939656781592, −17.09747529554943937637557945610, −16.38771542332458198357336366429, −15.28275319335035620762996120674, −14.95774416581407897893054051556, −14.037768163972057041363267490546, −13.49847656275560619290706027691, −12.737941273674591808959131682181, −12.16306466765195739327799336636, −11.77938351847572501371700711765, −10.911886759721275692293884433907, −10.18093692844339960624475577029, −9.672361981921943027770535347416, −8.53224464862447728593482746848, −7.66330342238982325876482614429, −6.65774774066857338724266408511, −6.238033836918598813674392867351, −5.56812860097789837552177428536, −4.91825003169561029466076326611, −4.41755252281275935623267945367, −3.04837902292093887928447202956, −2.23023842891665026905691476567, −1.70526459340940983511931701170, −0.897962106597933265052303399023, 1.14426982170156944094862924612, 1.79745385325058588270504024466, 3.07116848912829735228782539054, 3.724464972535009017035783178059, 4.67913739831430082195433266309, 5.03747622145743472157646665007, 5.77963576964972859180562176505, 6.66399841043722598841036900685, 6.937842665848049986117883865758, 7.887320784349397914293068569123, 9.138520108641238703750208661532, 9.64580417306406064183368962401, 10.52232974201398713307708558592, 11.20189231781785404414323940788, 11.79812571594994342293867440493, 12.25514784369854924384744363949, 13.549041596035528743092532098284, 13.77107992146634290992924970283, 14.450320660889680019292228117873, 15.04452546911252108811306768368, 16.09344687690125842762145663620, 16.56552381028151277476656527252, 17.03203212741139510886735874378, 17.70101170022926573688933657602, 18.19510877402173176459944134781

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