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Properties

Label 1-185-185.87-r0-0-0
Degree 11
Conductor 185185
Sign 0.959−0.283i0.959 - 0.283i
Analytic cond. 0.8591360.859136
Root an. cond. 0.8591360.859136
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  + (−0.939 − 0.342i)2-s + (−0.342 − 0.939i)3-s + (0.766 + 0.642i)4-s + i·6-s + (0.984 + 0.173i)7-s + (−0.5 − 0.866i)8-s + (−0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (0.342 − 0.939i)12-s + (0.766 + 0.642i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (0.939 − 0.342i)18-s + (0.342 + 0.939i)19-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.342 − 0.939i)3-s + (0.766 + 0.642i)4-s + i·6-s + (0.984 + 0.173i)7-s + (−0.5 − 0.866i)8-s + (−0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (0.342 − 0.939i)12-s + (0.766 + 0.642i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (0.939 − 0.342i)18-s + (0.342 + 0.939i)19-s + ⋯

Functional equation

Λ(s)=(185s/2 ΓR(s) L(s)=((0.959−0.283i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(185s/2 ΓR(s) L(s)=((0.959−0.283i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 185185    =    5⋅375 \cdot 37
Sign: 0.959−0.283i0.959 - 0.283i
Analytic conductor: 0.8591360.859136
Root analytic conductor: 0.8591360.859136
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ185(87,⋅)\chi_{185} (87, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 185, (0: ), 0.959−0.283i)(1,\ 185,\ (0:\ ),\ 0.959 - 0.283i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.7341742005−0.1061136530i0.7341742005 - 0.1061136530i
L(12)L(\frac12) ≈\approx 0.7341742005−0.1061136530i0.7341742005 - 0.1061136530i
L(1)L(1) ≈\approx 0.7079088913−0.1511617112i0.7079088913 - 0.1511617112i
L(1)L(1) ≈\approx 0.7079088913−0.1511617112i0.7079088913 - 0.1511617112i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
37 1 1
good2 1+(−0.939−0.342i)T 1 + (-0.939 - 0.342i)T
3 1+(−0.342−0.939i)T 1 + (-0.342 - 0.939i)T
7 1+(0.984+0.173i)T 1 + (0.984 + 0.173i)T
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
17 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
19 1+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
23 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
29 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
31 1−iT 1 - iT
41 1+(−0.766−0.642i)T 1 + (-0.766 - 0.642i)T
43 1+T 1 + T
47 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
53 1+(0.984−0.173i)T 1 + (0.984 - 0.173i)T
59 1+(0.984−0.173i)T 1 + (0.984 - 0.173i)T
61 1+(0.642−0.766i)T 1 + (0.642 - 0.766i)T
67 1+(−0.984−0.173i)T 1 + (-0.984 - 0.173i)T
71 1+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
73 1+iT 1 + iT
79 1+(−0.984−0.173i)T 1 + (-0.984 - 0.173i)T
83 1+(−0.642−0.766i)T 1 + (-0.642 - 0.766i)T
89 1+(−0.984+0.173i)T 1 + (-0.984 + 0.173i)T
97 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−27.0942569713600301735431169453, −26.719034444042492763850204724467, −25.54130323810650106403554675768, −24.4829683704038516164888159883, −23.66409536741845191259206785001, −22.441577460759229402644123004086, −21.32489880196805398381340007526, −20.45340336635528262076798042233, −19.67206916930591311432200307045, −18.06380515117943460982484296807, −17.69740198854951780352587876533, −16.45932179786169968832251510709, −15.83652355657665594749489560863, −14.79459231951228589878524693058, −13.86024075441835493197565998006, −11.75490351555692061401639493748, −11.0548745042781213548155466381, −10.308680195364141443545281429551, −8.91402492219535355503475209614, −8.403390033352109993823491176432, −6.82635311134441417151694069330, −5.68360033929736579209624103604, −4.58647778750504902023255348707, −2.91716736668036147019791984446, −0.960848802261530792490641096469, 1.44003285423419141366087450649, 2.11995458240231623536888494397, 4.06910944566146819185172417206, 5.88481323224742391195312564130, 6.97060679304233829922199528551, 7.96921318883155407597717136010, 8.79957310952759427179230912592, 10.20434400779253811317326218095, 11.476682986411664070630338441809, 11.85842335767032667779844272542, 13.08873943017253620833004023201, 14.340749681511559554103192832403, 15.650492958436297019913019256840, 16.95871417351690107007005732366, 17.64946367003780750204507177039, 18.344805491751151422539836858530, 19.28401622753642328464296379632, 20.232376698771394456548533283888, 21.17457383819596444772311714774, 22.34472099389303815412798039445, 23.615312193229608808539167003099, 24.47946626337686920111083400086, 25.28769798756379712866340435531, 26.17083029279790511829386895805, 27.462416942671377913102146377923

Graph of the ZZ-function along the critical line