Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.695 - 0.718i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.173 − 0.984i)2-s + (0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (−0.116 − 0.993i)5-s + (0.686 − 0.727i)6-s + (0.597 + 0.802i)7-s + (−0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (−0.998 − 0.0581i)10-s + (0.998 − 0.0581i)11-s + (−0.597 − 0.802i)12-s + (0.597 + 0.802i)13-s + (0.893 − 0.448i)14-s + (0.448 − 0.893i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s,χ)  = 1  + (0.173 − 0.984i)2-s + (0.835 + 0.549i)3-s + (−0.939 − 0.342i)4-s + (−0.116 − 0.993i)5-s + (0.686 − 0.727i)6-s + (0.597 + 0.802i)7-s + (−0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (−0.998 − 0.0581i)10-s + (0.998 − 0.0581i)11-s + (−0.597 − 0.802i)12-s + (0.597 + 0.802i)13-s + (0.893 − 0.448i)14-s + (0.448 − 0.893i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.695 - 0.718i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.695 - 0.718i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.695 - 0.718i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (3501, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.695 - 0.718i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.594263230 - 1.099157822i$
$L(\frac12,\chi)$  $\approx$  $2.594263230 - 1.099157822i$
$L(\chi,1)$  $\approx$  1.497223559 - 0.5409809464i
$L(1,\chi)$  $\approx$  1.497223559 - 0.5409809464i

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

−18.24113878957552153186425203375, −17.74068610911141825263502817387, −17.586072633365740119141575634146, −16.398749253626188678604110860636, −15.63125620515071019857009870077, −14.93864184932359576420839918103, −14.555443699959840206953509888587, −13.93769060439781769373687703942, −13.35687786736134781308862638233, −12.80583887068267472811689161324, −11.66811468864751042271065999233, −11.071842732193035586655663931, −10.05534850777987751290133333582, −9.3713846340407847086412031017, −8.56985250555497047408590511590, −7.878278350555926126195823943922, −7.388409643958015281274685588849, −6.69877542781178387571480310262, −6.25308402976639926498786512583, −5.178213498027931196254885843802, −4.07920184897601647726477194457, −3.64901406269107502590135586884, −2.95948142845620575479971806902, −1.70830363620043188414396149298, −0.86121004803264331542306104207, 0.91549877390654183471064435230, 1.86647430886132737925843451448, 2.241899063453356496101071077918, 3.40907639250371145909686116534, 4.07974919285256438159718819743, 4.60267085160014485826858185824, 5.29117807070935628877789994886, 6.139433374427288432510146422285, 7.46275403247732548646673934647, 8.47228207752821314633642739442, 8.84592544320019412366315954671, 9.187880589746884133851120131494, 9.97483289614651043821405354619, 10.878559454098325605021948571274, 11.71334611351064817156490130569, 12.00743486449736446277276002528, 12.87135243154807484060064127577, 13.75793815025200047635082261825, 14.07351564965374513437575509863, 14.80168477514204443742169082191, 15.603735606017027295147750254402, 16.29127840330759179775454680034, 16.97135089584959109067272619358, 17.98280293362880138626169774853, 18.51159289917907900438223409191

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