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Properties

Label 1-2675-2675.2504-r1-0-0
Degree 11
Conductor 26752675
Sign 0.623−0.781i0.623 - 0.781i
Analytic cond. 287.468287.468
Root an. cond. 287.468287.468
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  + (−0.553 − 0.832i)2-s + (−0.523 − 0.851i)3-s + (−0.386 + 0.922i)4-s + (−0.419 + 0.907i)6-s + (−0.915 + 0.403i)7-s + (0.982 − 0.188i)8-s + (−0.451 + 0.892i)9-s + (0.563 − 0.826i)11-s + (0.988 − 0.153i)12-s + (−0.353 − 0.935i)13-s + (0.842 + 0.538i)14-s + (−0.700 − 0.713i)16-s + (−0.933 + 0.359i)17-s + (0.992 − 0.118i)18-s + (0.217 + 0.976i)19-s + ⋯
L(s)  = 1  + (−0.553 − 0.832i)2-s + (−0.523 − 0.851i)3-s + (−0.386 + 0.922i)4-s + (−0.419 + 0.907i)6-s + (−0.915 + 0.403i)7-s + (0.982 − 0.188i)8-s + (−0.451 + 0.892i)9-s + (0.563 − 0.826i)11-s + (0.988 − 0.153i)12-s + (−0.353 − 0.935i)13-s + (0.842 + 0.538i)14-s + (−0.700 − 0.713i)16-s + (−0.933 + 0.359i)17-s + (0.992 − 0.118i)18-s + (0.217 + 0.976i)19-s + ⋯

Functional equation

Λ(s)=(2675s/2 ΓR(s+1) L(s)=((0.623−0.781i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2675s/2 ΓR(s+1) L(s)=((0.623−0.781i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 26752675    =    52⋅1075^{2} \cdot 107
Sign: 0.623−0.781i0.623 - 0.781i
Analytic conductor: 287.468287.468
Root analytic conductor: 287.468287.468
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2675(2504,⋅)\chi_{2675} (2504, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2675, (1: ), 0.623−0.781i)(1,\ 2675,\ (1:\ ),\ 0.623 - 0.781i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.4267894595−0.2054848712i0.4267894595 - 0.2054848712i
L(12)L(\frac12) ≈\approx 0.4267894595−0.2054848712i0.4267894595 - 0.2054848712i
L(1)L(1) ≈\approx 0.4269796785−0.2927996891i0.4269796785 - 0.2927996891i
L(1)L(1) ≈\approx 0.4269796785−0.2927996891i0.4269796785 - 0.2927996891i

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
107 1 1
good2 1+(−0.553−0.832i)T 1 + (-0.553 - 0.832i)T
3 1+(−0.523−0.851i)T 1 + (-0.523 - 0.851i)T
7 1+(−0.915+0.403i)T 1 + (-0.915 + 0.403i)T
11 1+(0.563−0.826i)T 1 + (0.563 - 0.826i)T
13 1+(−0.353−0.935i)T 1 + (-0.353 - 0.935i)T
17 1+(−0.933+0.359i)T 1 + (-0.933 + 0.359i)T
19 1+(0.217+0.976i)T 1 + (0.217 + 0.976i)T
23 1+(−0.772−0.634i)T 1 + (-0.772 - 0.634i)T
29 1+(0.997+0.0710i)T 1 + (0.997 + 0.0710i)T
31 1+(0.472−0.881i)T 1 + (0.472 - 0.881i)T
37 1+(0.949+0.314i)T 1 + (0.949 + 0.314i)T
41 1+(−0.513−0.858i)T 1 + (-0.513 - 0.858i)T
43 1+(−0.533+0.845i)T 1 + (-0.533 + 0.845i)T
47 1+(−0.919+0.392i)T 1 + (-0.919 + 0.392i)T
53 1+(−0.620−0.783i)T 1 + (-0.620 - 0.783i)T
59 1+(0.848+0.528i)T 1 + (0.848 + 0.528i)T
61 1+(0.503+0.864i)T 1 + (0.503 + 0.864i)T
67 1+(−0.683−0.729i)T 1 + (-0.683 - 0.729i)T
71 1+(−0.0769−0.997i)T 1 + (-0.0769 - 0.997i)T
73 1+(−0.297+0.954i)T 1 + (-0.297 + 0.954i)T
79 1+(−0.0177−0.999i)T 1 + (-0.0177 - 0.999i)T
83 1+(−0.910+0.413i)T 1 + (-0.910 + 0.413i)T
89 1+(−0.872−0.487i)T 1 + (-0.872 - 0.487i)T
97 1+(0.867−0.498i)T 1 + (0.867 - 0.498i)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

−19.34359399733334801289206804550, −18.187033832616679313266536523182, −17.62350935272501605606306041320, −17.074760312271598778547938473937, −16.36720470885480249517288583371, −15.79006070561424633675456948745, −15.34120832163343093219070049859, −14.40968474450328914273006745082, −13.822912205287980696908371546006, −12.91551908401201607203283825634, −11.82354364126824128905102832796, −11.27230717360322166771115521193, −10.212359887255533601147440263528, −9.78308425083398135933482578576, −9.24131153913403007724373260472, −8.549799411682723107886240855907, −7.26679372494661255243886834735, −6.67026940014819943291981677535, −6.30571004616402026266728520669, −5.09816938916196348676691795152, −4.54118351728158366670019491616, −3.8802316471381594194773485259, −2.62292462982480686114017439753, −1.351941747264584257317277402238, −0.2332351421002541607287586750, 0.42964944174438042222393334399, 1.30007007660324971257150880733, 2.32482123615361360883001100622, 2.95326938531653997043068084212, 3.854284110285990896095952166715, 4.908130570401207982384748362518, 6.08236281207429612733854465920, 6.39176094851925996410615305403, 7.4975737240118686649319342603, 8.27948675083259607343308685557, 8.734237646607416014975881689395, 9.91474642745277669698475582941, 10.30031795399715139664122765533, 11.31344777700270320637470896759, 11.82913380910544661892505418184, 12.53386697901714354440899644870, 13.07014342040723657162392934650, 13.673096652225812648749123544024, 14.60929478950147317875163812317, 15.848381929160472750566187440195, 16.45760949258597768710043206794, 17.05444903356206755907797216478, 17.86316465736268987755224149294, 18.350100215083095304410209090145, 19.08512440379817810322584711911

Graph of the ZZ-function along the critical line