↑

Properties

Label 1-41-41.13-r1-0-0
Degree $1$
Conductor $41$
Sign $-0.169 + 0.985i$
Analytic cond. $4.40606$
Root an. cond. $4.40606$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $-0.169 + 0.985i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ -0.169 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.082215447 + 1.283933712i\)
\(L(\frac12)\) \(\approx\) \(1.082215447 + 1.283933712i\)
\(L(1)\) \(\approx\) \(1.108412479 + 0.6605077861i\)
\(L(1)\) \(\approx\) \(1.108412479 + 0.6605077861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (0.156 + 0.987i)T \)
11 \( 1 + (-0.453 + 0.891i)T \)
13 \( 1 + (0.987 + 0.156i)T \)
17 \( 1 + (-0.891 - 0.453i)T \)
19 \( 1 + (0.987 - 0.156i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (-0.891 + 0.453i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (0.156 - 0.987i)T \)
53 \( 1 + (0.891 - 0.453i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (0.453 + 0.891i)T \)
71 \( 1 + (0.453 - 0.891i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (0.156 + 0.987i)T \)
97 \( 1 + (-0.453 - 0.891i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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