Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.892 + 0.450i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (0.993 − 0.116i)3-s + 4-s + (−0.727 − 0.686i)5-s + (−0.993 + 0.116i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (0.727 + 0.686i)10-s + (−0.727 + 0.686i)11-s + (0.993 − 0.116i)12-s + (0.686 − 0.727i)13-s + (−0.973 − 0.230i)14-s + (−0.802 − 0.597i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (0.993 − 0.116i)3-s + 4-s + (−0.727 − 0.686i)5-s + (−0.993 + 0.116i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (0.727 + 0.686i)10-s + (−0.727 + 0.686i)11-s + (0.993 − 0.116i)12-s + (0.686 − 0.727i)13-s + (−0.973 − 0.230i)14-s + (−0.802 − 0.597i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.892 + 0.450i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.892 + 0.450i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.892 + 0.450i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1863, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.892 + 0.450i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.595375980 + 0.3794811968i$
$L(\frac12,\chi)$  $\approx$  $1.595375980 + 0.3794811968i$
$L(\chi,1)$  $\approx$  1.022198825 + 0.01667366007i
$L(1,\chi)$  $\approx$  1.022198825 + 0.01667366007i

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

−18.58986430982770280028624729244, −17.981310467721119013769522530870, −17.15205491680549974238004026, −16.23229397853062672165136417496, −15.6323202656958287523692267806, −15.26565985622131255140766083394, −14.3793744356447617963682608905, −13.93022495288143953443055248015, −13.02001207236441204858584789793, −11.93995421048477893763941523757, −11.32010947491320517865136115727, −10.67846662430035522497225916806, −10.2821810716897963120866744582, −9.17467309110491012499379982511, −8.568006562801795631305848332738, −8.02294494792653315520542556408, −7.58122268041551133010820848559, −6.77216585557842939638729925327, −6.02709887120406389523636991084, −4.70856092327279723551609323893, −3.934710821364506820210217358915, −3.14945036692902681074973550236, −2.41700262249955746928104197969, −1.69174875737245803793284018103, −0.61464351296650461806800478865, 1.12181860538100295121840946348, 1.41317217260565525385868089732, 2.61928972898232895623315144289, 3.11398017998405014586004700072, 4.21372884386777488148118891478, 5.00317898019157714278226998868, 5.827809793153178300380313362750, 7.23745921248410274425678694278, 7.520813731453323915401513650685, 8.11653571470983638278499299338, 8.679822870224359428244991696843, 9.35418811543382906726605472423, 9.984537021202873564897898577807, 10.97358995836437574026940808341, 11.5029052420717343702440279546, 12.32370071868319929813705368023, 12.93704062666662728020042937849, 13.773393145371748567711048537713, 14.72334293295474347568621529382, 15.261519278121519966399448879, 15.98824008784132288921767395915, 16.09242346574242639772447342500, 17.527405897394371376570760866471, 17.904620343534634544121184338198, 18.43691568834578848427757465389

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