Properties

Label 1-2664-2664.1285-r0-0-0
Degree 11
Conductor 26642664
Sign 0.5460.837i0.546 - 0.837i
Analytic cond. 12.371512.3715
Root an. cond. 12.371512.3715
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  + (−0.5 − 0.866i)5-s + 7-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)35-s + (−0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + 7-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)35-s + (−0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + 49-s + ⋯

Functional equation

Λ(s)=(2664s/2ΓR(s)L(s)=((0.5460.837i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2664s/2ΓR(s)L(s)=((0.5460.837i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 26642664    =    2332372^{3} \cdot 3^{2} \cdot 37
Sign: 0.5460.837i0.546 - 0.837i
Analytic conductor: 12.371512.3715
Root analytic conductor: 12.371512.3715
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2664(1285,)\chi_{2664} (1285, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2664, (0: ), 0.5460.837i)(1,\ 2664,\ (0:\ ),\ 0.546 - 0.837i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.4576260770.7890672285i1.457626077 - 0.7890672285i
L(12)L(\frac12) \approx 1.4576260770.7890672285i1.457626077 - 0.7890672285i
L(1)L(1) \approx 1.0962152560.2048777902i1.096215256 - 0.2048777902i
L(1)L(1) \approx 1.0962152560.2048777902i1.096215256 - 0.2048777902i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
37 1 1
good5 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
7 1+T 1 + T
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
17 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
19 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
23 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
29 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
31 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
41 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
43 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
47 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
53 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
59 1+T 1 + T
61 1+T 1 + T
67 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
71 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
73 1+T 1 + T
79 1T 1 - T
83 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
89 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
97 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
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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

−19.47410888950860678803480812826, −18.70618881600275075057151380799, −18.10461891154492282900483525813, −17.375950733247536750922878031799, −16.59260703866024542785703825107, −15.90004442626215544135814718876, −15.006000239770370510026622952, −14.44046236891357522853839966686, −14.01361731538217611430883715027, −13.11883473518666955639697869854, −11.94930353022926085965930956341, −11.39794475517774356209289362264, −11.14711001095838568347539321386, −10.121267793986835122647825221509, −9.23455674083039747910069983434, −8.504401229565311918449444184765, −7.69752038227246417754076241636, −6.9837325366079494344040204405, −6.419175028789577480406622698683, −5.20743579672021014682409048373, −4.656529942758767622531234639170, −3.615281535465638204751105923446, −2.959118168338840350904663359266, −1.97531602723750407206100912627, −0.95686974346214562499056117877, 0.63789797373158624707446976709, 1.632906770945987667450552023835, 2.348602837773391860515740699015, 3.8166861655805678064275745860, 4.21853378266099915180718055178, 5.14409497992975977877157437684, 5.68667862091401648041552327232, 6.87074544755388236563791820820, 7.73949641661833404066876097402, 8.23804258292088408052015560647, 8.85302512066659360341024046431, 9.954330218681768039354052004474, 10.43725514848050182485002444256, 11.53624510712038160315748308148, 12.06732592837531866466821898781, 12.67150712767393149183878138909, 13.35264256230506032151043875955, 14.5199732149413899050537341207, 14.9437296763752736084027116965, 15.46647945664166659805866434616, 16.60268254762613048769456512013, 17.2018075699164555370103525767, 17.45476194295152766220231650454, 18.6370079781438830473018195227, 19.164748995222975205759992482

Graph of the ZZ-function along the critical line