Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.923 + 0.383i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (0.597 − 0.802i)3-s + 4-s + (−0.973 + 0.230i)5-s + (−0.597 + 0.802i)6-s + (−0.286 + 0.957i)7-s − 8-s + (−0.286 − 0.957i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.597 − 0.802i)12-s + (−0.973 + 0.230i)13-s + (0.286 − 0.957i)14-s + (−0.396 + 0.918i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (0.597 − 0.802i)3-s + 4-s + (−0.973 + 0.230i)5-s + (−0.597 + 0.802i)6-s + (−0.286 + 0.957i)7-s − 8-s + (−0.286 − 0.957i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.597 − 0.802i)12-s + (−0.973 + 0.230i)13-s + (0.286 − 0.957i)14-s + (−0.396 + 0.918i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.923 + 0.383i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.923 + 0.383i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.923 + 0.383i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1656, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.923 + 0.383i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9172103622 + 0.1828230784i$
$L(\frac12,\chi)$  $\approx$  $0.9172103622 + 0.1828230784i$
$L(\chi,1)$  $\approx$  0.7099148549 - 0.05986418758i
$L(1,\chi)$  $\approx$  0.7099148549 - 0.05986418758i

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

−18.784058633144040036000453399541, −17.47632624538676668301775808417, −16.92938209825518480738039422443, −16.41521393569126734632987995161, −15.9758918013560060544073537093, −15.10623840005500009947929780575, −14.51043299721318079690339928714, −14.037106467445378845342021322329, −12.62000662058122515086355363870, −12.250307339408865631996843076997, −11.20585741725332127154725188906, −10.7412185094479723825309688138, −10.02854153393782262678951808901, −9.43533445772458318847813943338, −8.71223291065759505925074753491, −8.06613756843017210310362147100, −7.4201736677157635783496154236, −6.90234207754058900812284115978, −5.75038858787784020792809082192, −4.79110366015774631753593293065, −3.68894631610049154683839057349, −3.6379807410225365309813411005, −2.54438356963877346098261765921, −1.43935838321818815042124097922, −0.47657473920413877154704914735, 0.7692659099685992474771930420, 1.638133786915346997777001293999, 2.64845671887331638323206085086, 3.01513874686010165009292027400, 3.93803099398399698979415831001, 5.22999235854749890332985750929, 6.18611029772430131933399646129, 6.89814017438482351640363194051, 7.50109056158558003064856136069, 7.90005342946789736091569127485, 8.84112271666320929690256463503, 9.47128364067636187904084046948, 9.72411452617934370408936681055, 11.25358052499259231693288806144, 11.65364725298419936628563950350, 12.189700390619231945015384194413, 12.65662943665620934294835287640, 13.96305688562479377212635348810, 14.58946374602986058889252437886, 15.15980769098859537353733476696, 15.843257167341841137662973419183, 16.415479132489645756732729319701, 17.48524207438041659572941264313, 17.86783902021494977740502223211, 18.81576660662962378201056843097

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