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  :  $\chi_{4033} (701, \cdot )$
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

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

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