Properties

Label 1-520-520.493-r1-0-0
Degree 11
Conductor 520520
Sign 0.850+0.525i-0.850 + 0.525i
Analytic cond. 55.881755.8817
Root an. cond. 55.881755.8817
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + i·7-s − 9-s + 11-s i·17-s − 19-s + 21-s + i·23-s + i·27-s + 29-s − 31-s i·33-s i·37-s − 41-s i·43-s + ⋯
L(s)  = 1  i·3-s + i·7-s − 9-s + 11-s i·17-s − 19-s + 21-s + i·23-s + i·27-s + 29-s − 31-s i·33-s i·37-s − 41-s i·43-s + ⋯

Functional equation

Λ(s)=(520s/2ΓR(s+1)L(s)=((0.850+0.525i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(520s/2ΓR(s+1)L(s)=((0.850+0.525i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 520520    =    235132^{3} \cdot 5 \cdot 13
Sign: 0.850+0.525i-0.850 + 0.525i
Analytic conductor: 55.881755.8817
Root analytic conductor: 55.881755.8817
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ520(493,)\chi_{520} (493, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 520, (1: ), 0.850+0.525i)(1,\ 520,\ (1:\ ),\ -0.850 + 0.525i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.019301392660.06794373990i0.01930139266 - 0.06794373990i
L(12)L(\frac12) \approx 0.019301392660.06794373990i0.01930139266 - 0.06794373990i
L(1)L(1) \approx 0.84801231260.2001885515i0.8480123126 - 0.2001885515i
L(1)L(1) \approx 0.84801231260.2001885515i0.8480123126 - 0.2001885515i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
13 1 1
good3 1+T 1 + T
7 1iT 1 - iT
11 1 1
17 1 1
19 1+iT 1 + iT
23 1 1
29 1T 1 - T
31 1 1
37 1+T 1 + T
41 1 1
43 1 1
47 1 1
53 1 1
59 1 1
61 1iT 1 - iT
67 1 1
71 1T 1 - T
73 1 1
79 1+T 1 + T
83 1 1
89 1+iT 1 + iT
97 1 1
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−23.56218803276308000524481410951, −23.00492538527100197324209269156, −21.971086305800787782571113277775, −21.464765442764555908696234518357, −20.31840612311396798709513458692, −19.923128081188688595991809990167, −18.946172486106917322991229230647, −17.53810431449030372007993160239, −16.90591125439961488089495610285, −16.394746397736307090195233211229, −15.14600155069740803304946760773, −14.57410951031407983921122063622, −13.7275296936409919341543304530, −12.59333644712353988160682835280, −11.5099390177137716255848484271, −10.58436502027786263374803948811, −10.09418311986337203823996064975, −8.92473220832702080804350338515, −8.216172161163277057462121740285, −6.81439233338221299678401321338, −6.026551160440865668399088586225, −4.57899389776792031405443250448, −4.09665955017170492073809404139, −3.06143435230847750324050643443, −1.48378498217064249556033326121, 0.01735230950147015540040240873, 1.45633552525518536843757208962, 2.3713881735861690967443408574, 3.48120666243944729733399832592, 4.98951631022137095977471961425, 5.9869815085371620619738077486, 6.751515667655515034639847054659, 7.730657252379397089936852931009, 8.80725972413671467002348224740, 9.34359653656719836718119895229, 10.89425172352993276556532382311, 11.84621675824583380136748736478, 12.29653052955928705646995156961, 13.33045800773207979073688812662, 14.20847034031471918753003124793, 14.98107530401516279808912800761, 16.044153620186619589674954899086, 17.10524271783263660829155274176, 17.84489551938149370332448568582, 18.6404338334639499412801417873, 19.37414921225198071260376136809, 20.05285991863407142204632498253, 21.23832838138698756719441007555, 22.08991291743246108139949298546, 22.88921246451969237927829954438

Graph of the ZZ-function along the critical line