Properties

Degree 1
Conductor 41
Sign $-0.169 + 0.985i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.587 + 0.809i)2-s + (−0.707 − 0.707i)3-s + (−0.309 + 0.951i)4-s + (0.951 + 0.309i)5-s + (0.156 − 0.987i)6-s + (0.156 + 0.987i)7-s + (−0.951 + 0.309i)8-s + i·9-s + (0.309 + 0.951i)10-s + (−0.453 + 0.891i)11-s + (0.891 − 0.453i)12-s + (0.987 + 0.156i)13-s + (−0.707 + 0.707i)14-s + (−0.453 − 0.891i)15-s + (−0.809 − 0.587i)16-s + (−0.891 − 0.453i)17-s + ⋯
L(s,χ)  = 1  + (0.587 + 0.809i)2-s + (−0.707 − 0.707i)3-s + (−0.309 + 0.951i)4-s + (0.951 + 0.309i)5-s + (0.156 − 0.987i)6-s + (0.156 + 0.987i)7-s + (−0.951 + 0.309i)8-s + i·9-s + (0.309 + 0.951i)10-s + (−0.453 + 0.891i)11-s + (0.891 − 0.453i)12-s + (0.987 + 0.156i)13-s + (−0.707 + 0.707i)14-s + (−0.453 − 0.891i)15-s + (−0.809 − 0.587i)16-s + (−0.891 − 0.453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.169 + 0.985i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.169 + 0.985i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $-0.169 + 0.985i$
motivic weight  =  \(0\)
character  :  $\chi_{41} (13, \cdot )$
Sato-Tate  :  $\mu(40)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 41,\ (1:\ ),\ -0.169 + 0.985i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.082215447 + 1.283933712i$
$L(\frac12,\chi)$  $\approx$  $1.082215447 + 1.283933712i$
$L(\chi,1)$  $\approx$  1.108412479 + 0.6605077861i
$L(1,\chi)$  $\approx$  1.108412479 + 0.6605077861i

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

−33.73441855738938560120327008310, −33.000643146107811684391752062052, −32.28265412674473725026595433376, −30.6367062828933703939872258709, −29.27197616189864242922447929539, −28.83850988985284221338426594014, −27.472246772071718163051660487282, −26.312065886624406775539098859059, −24.27937278801016683833481186543, −23.25114997268292982369132740778, −22.04641681533851522219312411900, −21.08157748441101846405873835051, −20.282784672282964072281426001253, −18.34040476600215981607894811638, −17.13450623787367599949701556396, −15.712245843488231833496669036060, −13.967220053335036136896943865460, −13.05609496607579940198742094572, −11.21316141692938514089598155493, −10.47542384799660887987264782524, −9.14905432924210310689584361244, −6.18984872156243411274838112934, −5.05458395599090769021463482502, −3.53821930930227491327877659451, −1.057370782739822300545366586298, 2.35236702024620623106470566976, 5.08505435212995013135058937754, 6.080484323399934276644590446310, 7.2928461580653797589587750228, 9.076334129666277123366722673837, 11.25202889006392165710693924782, 12.69971311516937637993482670744, 13.59883188284805425760598115741, 15.10425919228552528163675215838, 16.48746360997833057158017202244, 18.00616151156401337842290313321, 18.25751002331967723814174286873, 20.87580220092835030513945163035, 22.151218761850800614905817405616, 22.89382718932884641419921629927, 24.35894781989499914786328056043, 25.11959507609796493897520788070, 26.16973969714327181806862104516, 28.101870649105953917735303136413, 29.14468572406075792804632221912, 30.548409484011857306161195023992, 31.248694503821280715044617699986, 33.14369468223177766526600415038, 33.64276273358387460967940403955, 34.83813006285512385778668058732

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