Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.363 - 0.931i$
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 + (0.835 + 0.549i)3-s + 4-s + (0.802 − 0.597i)5-s + (−0.835 − 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (−0.802 + 0.597i)10-s + (0.802 + 0.597i)11-s + (0.835 + 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.998 − 0.0581i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (0.835 + 0.549i)3-s + 4-s + (0.802 − 0.597i)5-s + (−0.835 − 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (−0.802 + 0.597i)10-s + (0.802 + 0.597i)11-s + (0.835 + 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.998 − 0.0581i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.363 - 0.931i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.363 - 0.931i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.363 - 0.931i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2238, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.363 - 0.931i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.586000868 - 1.084088324i$
$L(\frac12,\chi)$  $\approx$  $1.586000868 - 1.084088324i$
$L(\chi,1)$  $\approx$  1.137601201 - 0.1893135019i
$L(1,\chi)$  $\approx$  1.137601201 - 0.1893135019i

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

−18.85245514024578831866525083463, −17.909421291984652635615078723567, −17.48614375944114246076903823192, −16.82484392908321025194442661673, −15.899685688344834996419806472441, −14.93515038740516472946226262477, −14.675878860040403160717570799224, −14.102884254926168839165889828206, −13.053905669596695339908232857791, −12.41838762020568083154852394222, −11.51779363763631863265410266595, −11.12429499049855092706264648217, −9.97741503723011905219132070210, −9.4602977358500073895751297710, −8.90085941106679246787184379192, −8.31603800773824949242708914357, −7.52147961516092154026872314930, −6.729238306333782223057167044778, −6.21298495015537663686472405549, −5.56842588535738065493872704033, −4.0825955228480879750040625512, −3.13607739368191246929897630311, −2.470840315530448548666118321867, −1.74206016080242886388178349210, −1.33896139496400717045725884294, 0.63302169917191485358697388864, 1.45330743686252582264790900643, 2.38369724951651132407749722773, 2.85601536513770865648651431555, 4.14519614262008543241770749177, 4.71784137644022115028973951853, 5.56594924636423108628299061477, 6.77750877731429449735707761639, 7.26186673659825866794087237237, 7.98203151362699174780766618695, 8.98819101820851390982423526978, 9.10549509723039931729323366881, 9.98508404888491689816369179502, 10.42880296683729220530578971384, 11.155553992945588041689358209487, 12.085265464234067265196474113433, 13.00578135460357420453916284078, 13.58798880226441414927540052478, 14.433359690199571307770136884223, 14.98382440035003112255569690978, 15.683640118804384567863672043477, 16.51357895071710437024681776290, 17.06675975176150726020370913369, 17.54755580003324466001705847988, 18.13815941056127887563266421936

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