Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.644 - 0.764i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (−0.893 + 0.448i)3-s + 4-s + (0.116 + 0.993i)5-s + (0.893 − 0.448i)6-s + (0.597 + 0.802i)7-s − 8-s + (0.597 − 0.802i)9-s + (−0.116 − 0.993i)10-s + (0.116 − 0.993i)11-s + (−0.893 + 0.448i)12-s + (0.993 − 0.116i)13-s + (−0.597 − 0.802i)14-s + (−0.549 − 0.835i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (−0.893 + 0.448i)3-s + 4-s + (0.116 + 0.993i)5-s + (0.893 − 0.448i)6-s + (0.597 + 0.802i)7-s − 8-s + (0.597 − 0.802i)9-s + (−0.116 − 0.993i)10-s + (0.116 − 0.993i)11-s + (−0.893 + 0.448i)12-s + (0.993 − 0.116i)13-s + (−0.597 − 0.802i)14-s + (−0.549 − 0.835i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.644 - 0.764i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.644 - 0.764i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.644 - 0.764i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (166, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.644 - 0.764i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6421496024 - 0.2984834153i$
$L(\frac12,\chi)$  $\approx$  $0.6421496024 - 0.2984834153i$
$L(\chi,1)$  $\approx$  0.5963742196 + 0.07458450244i
$L(1,\chi)$  $\approx$  0.5963742196 + 0.07458450244i

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

−18.25432326787727313282512593707, −17.655932735854008016985241290869, −17.46139576027937765719824485327, −16.60495482500573407939918072296, −16.27821806207141997734970512199, −15.4681197747003104626762617650, −14.60023265080416228924101231485, −13.59024306067382570335338410754, −12.8645090058709199896182611111, −12.28192328218346561591280573258, −11.660473114859059520748419569945, −10.7253594069751607784926458138, −10.54232801488128775679508353122, −9.568899728061817822161991029287, −8.77184957610690653591382407532, −8.054865759284532878083669954030, −7.51279543999078926852670503903, −6.67765129848124956034232728366, −6.090437383408332250247676913914, −5.18433082927408803442088770770, −4.47025441520085534943653450026, −3.61026434926005581806043138623, −2.03956128477561549686927494545, −1.389310417913503915573232002929, −1.10399188739698432099552620124, 0.367773056915521501273589012557, 1.289548207626890802930635359231, 2.51079659893409777210444825133, 2.96008043943420613786221545043, 4.065438860887847443510287856777, 5.09768986133497699205303785637, 6.11347869407206173244876347710, 6.19421782877061575238521024706, 7.12732566105529551109278414320, 7.93875636933440300401159091432, 8.956729828511574224097221167672, 9.18670190633501389180965583407, 10.25413879436217331354769789115, 10.94329652983596207720387912998, 11.28813404698673133437428073533, 11.634562995036082899780123536, 12.676982908459856855121320835242, 13.66083306561208184104912753284, 14.61178155643646946442262949544, 15.21337372977473373950408559799, 15.932048816546128483987443393258, 16.15680809284228101289986996616, 17.37887266356695960019266640756, 17.65221435900321096648079086500, 18.26974179849501253349771396359

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