Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.0853 - 0.996i$
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 + (−0.993 + 0.116i)3-s + 4-s + (0.686 − 0.727i)5-s + (0.993 − 0.116i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.993 + 0.116i)12-s + (0.686 − 0.727i)13-s + (−0.973 − 0.230i)14-s + (−0.597 + 0.802i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (−0.993 + 0.116i)3-s + 4-s + (0.686 − 0.727i)5-s + (0.993 − 0.116i)6-s + (0.973 + 0.230i)7-s − 8-s + (0.973 − 0.230i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.993 + 0.116i)12-s + (0.686 − 0.727i)13-s + (−0.973 − 0.230i)14-s + (−0.597 + 0.802i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0853 - 0.996i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0853 - 0.996i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.0853 - 0.996i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1965, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.0853 - 0.996i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6447552941 - 0.7023551544i$
$L(\frac12,\chi)$  $\approx$  $0.6447552941 - 0.7023551544i$
$L(\chi,1)$  $\approx$  0.6632837893 - 0.1645069851i
$L(1,\chi)$  $\approx$  0.6632837893 - 0.1645069851i

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

−18.44838122969047349898641556808, −18.01173505787499843383858923347, −17.458103715389573632194258900989, −16.85834769701908060637292136078, −16.105639677878257376921704402569, −15.46372135008931753393096413527, −14.73508348821299181192856774033, −13.85878373951446339299838419434, −13.18964243922912386532571859633, −12.14089224515705461351951937173, −11.340121868431780495470916857450, −11.20521855624626352561689043976, −10.42668462423129441457878638018, −9.5709488276282195604582988416, −9.37023927796073841114235564487, −7.82558756698802734847264462050, −7.53824443663095969708053926778, −6.894570577886592640460243111520, −6.023509993707631686776125696505, −5.44715972023637473088552239152, −4.66063639274330354968310105301, −3.459069692201219622262601662179, −2.3953814237513392751512283791, −1.66641032107972039053181912593, −1.06429430478032474214257515696, 0.48886554332486067059672860874, 1.29283176770117969629235622375, 1.842283266770104891783150494958, 2.9974989393827371929827478313, 4.077394464989298815743287798584, 5.240121627590849321858528885160, 5.638730902000529015532556217632, 6.08612599890367689105028940555, 7.18393540531386400126542942827, 8.07366761552890415976841145276, 8.44213342591296999661338412507, 9.29746063687605643660882340989, 10.15589600663302519570446929444, 10.68235255228187023366645231786, 11.11566408049524872328691064026, 12.11488962764556623588554288867, 12.479168529608333982122234765, 13.362173360520397098881309483002, 14.281141081106930984818250891342, 15.21340154152239944280500203509, 15.8750983404001431809119045524, 16.48077758353043592106753848880, 16.962802646891278657223185414794, 17.68764350938508364166921743641, 18.22134557479008574081569394299

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