Properties

Label 1-185-185.47-r1-0-0
Degree 11
Conductor 185185
Sign 0.6440.764i0.644 - 0.764i
Analytic cond. 19.881019.8810
Root an. cond. 19.881019.8810
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  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s + (−0.866 − 0.5i)7-s i·8-s + (0.5 + 0.866i)9-s + 11-s + (0.866 − 0.5i)12-s + (0.866 + 0.5i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s + (−0.866 − 0.5i)7-s i·8-s + (0.5 + 0.866i)9-s + 11-s + (0.866 − 0.5i)12-s + (0.866 + 0.5i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

Λ(s)=(185s/2ΓR(s+1)L(s)=((0.6440.764i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(185s/2ΓR(s+1)L(s)=((0.6440.764i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 185185    =    5375 \cdot 37
Sign: 0.6440.764i0.644 - 0.764i
Analytic conductor: 19.881019.8810
Root analytic conductor: 19.881019.8810
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ185(47,)\chi_{185} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 185, (1: ), 0.6440.764i)(1,\ 185,\ (1:\ ),\ 0.644 - 0.764i)

Particular Values

L(12)L(\frac{1}{2}) \approx 3.8759921151.801404158i3.875992115 - 1.801404158i
L(12)L(\frac12) \approx 3.8759921151.801404158i3.875992115 - 1.801404158i
L(1)L(1) \approx 2.2724926800.6408059560i2.272492680 - 0.6408059560i
L(1)L(1) \approx 2.2724926800.6408059560i2.272492680 - 0.6408059560i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
37 1 1
good2 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
3 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
7 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
11 1+T 1 + T
13 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
17 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
19 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
23 1iT 1 - iT
29 1T 1 - T
31 1+T 1 + T
41 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
43 1iT 1 - iT
47 1+iT 1 + iT
53 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
59 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
61 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
67 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
71 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
73 1iT 1 - iT
79 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
83 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
89 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
97 1+iT 1 + iT
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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

−26.72082975511269388788276523433, −25.61795081194035479807986037581, −25.303106059733753203150855030530, −24.41177815890740547204473743702, −23.28905461877097229934251683314, −22.510551196936095943057988204723, −21.397588002544612846044779263856, −20.48730816343648327731117840560, −19.50224155168371676819341327674, −18.54092253368550393821695231752, −17.2309593898328085161842863875, −16.06209896741638570017752220385, −15.19603730965151302936457658557, −14.28795454086290985936878119973, −13.37659127331363198395426256591, −12.53748475326376151029584332346, −11.67700407535228486058130936984, −9.8056813014266473185788402915, −8.65229193918472378844959175673, −7.67715961773922533101882710214, −6.50177132890227714893185318017, −5.70046445253553867008226375149, −3.738541196627590531902666385335, −3.24072786253104252002218783458, −1.61993516509753482632350990938, 1.20134797481881114648601274575, 2.82049594341687468815554555004, 3.69515324847073128294684559911, 4.608193522282537551929049323528, 6.19622229551611807440014522972, 7.26060744371695802824067612867, 9.01383298558352160423664736529, 9.797470919408957046045482185658, 10.84541208231910676706980063454, 12.02682165581028253916214873626, 13.31308705604730998822071712718, 13.89895896128479385301374689105, 14.8301941283994161144045961130, 15.92988545111724298012316185896, 16.64911719043803901919544880160, 18.68370945299884975111557083407, 19.41745997611607047823260453202, 20.299211503241279928607447108023, 20.95402642002816193238180711624, 22.12873053819101222180299920903, 22.704699135636709030392380340805, 23.90868652277994579812713154829, 24.96705955159777559605389967001, 25.7754978819631957054914622379, 26.79946702661941151262149677047

Graph of the ZZ-function along the critical line