Properties

Label 1-675-675.139-r0-0-0
Degree 11
Conductor 675675
Sign 0.009300.999i0.00930 - 0.999i
Analytic cond. 3.134683.13468
Root an. cond. 3.134683.13468
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  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 + 0.984i)7-s + (0.104 + 0.994i)8-s + (0.0348 + 0.999i)11-s + (−0.848 − 0.529i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.913 − 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.559 − 0.829i)22-s + (−0.990 − 0.139i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.241 − 0.970i)29-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 + 0.984i)7-s + (0.104 + 0.994i)8-s + (0.0348 + 0.999i)11-s + (−0.848 − 0.529i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.913 − 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.559 − 0.829i)22-s + (−0.990 − 0.139i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.241 − 0.970i)29-s + ⋯

Functional equation

Λ(s)=(675s/2ΓR(s)L(s)=((0.009300.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00930 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(675s/2ΓR(s)L(s)=((0.009300.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00930 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 675675    =    33523^{3} \cdot 5^{2}
Sign: 0.009300.999i0.00930 - 0.999i
Analytic conductor: 3.134683.13468
Root analytic conductor: 3.134683.13468
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ675(139,)\chi_{675} (139, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 675, (0: ), 0.009300.999i)(1,\ 675,\ (0:\ ),\ 0.00930 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.20128272030.1994177621i0.2012827203 - 0.1994177621i
L(12)L(\frac12) \approx 0.20128272030.1994177621i0.2012827203 - 0.1994177621i
L(1)L(1) \approx 0.5386109102+0.1028584145i0.5386109102 + 0.1028584145i
L(1)L(1) \approx 0.5386109102+0.1028584145i0.5386109102 + 0.1028584145i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1 1
good2 1+(0.848+0.529i)T 1 + (-0.848 + 0.529i)T
7 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
11 1+(0.0348+0.999i)T 1 + (0.0348 + 0.999i)T
13 1+(0.8480.529i)T 1 + (-0.848 - 0.529i)T
17 1+(0.9130.406i)T 1 + (-0.913 - 0.406i)T
19 1+(0.1040.994i)T 1 + (-0.104 - 0.994i)T
23 1+(0.9900.139i)T 1 + (-0.990 - 0.139i)T
29 1+(0.2410.970i)T 1 + (-0.241 - 0.970i)T
31 1+(0.7190.694i)T 1 + (-0.719 - 0.694i)T
37 1+(0.669+0.743i)T 1 + (-0.669 + 0.743i)T
41 1+(0.848+0.529i)T 1 + (0.848 + 0.529i)T
43 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
47 1+(0.7190.694i)T 1 + (0.719 - 0.694i)T
53 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
59 1+(0.03480.999i)T 1 + (0.0348 - 0.999i)T
61 1+(0.8820.469i)T 1 + (-0.882 - 0.469i)T
67 1+(0.2410.970i)T 1 + (0.241 - 0.970i)T
71 1+(0.104+0.994i)T 1 + (-0.104 + 0.994i)T
73 1+(0.6690.743i)T 1 + (-0.669 - 0.743i)T
79 1+(0.2410.970i)T 1 + (-0.241 - 0.970i)T
83 1+(0.997+0.0697i)T 1 + (0.997 + 0.0697i)T
89 1+(0.669+0.743i)T 1 + (0.669 + 0.743i)T
97 1+(0.9610.275i)T 1 + (-0.961 - 0.275i)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

−22.759652246193054108200355833045, −21.92307193171473408606510958885, −21.249499971563297800247346311365, −20.30019249451425104109596555712, −19.60357422343143794412998072117, −19.080259820109606613932026912096, −18.00211668828549531815745463907, −17.31336186843605256902084821802, −16.4121915609024750023981196715, −16.05819347956319751553487876925, −14.54457061498612405079931082650, −13.76214058354000158154230726116, −12.755775034915239387687767522371, −11.98917713189856537895219054118, −10.869323686444185828085166381472, −10.50027406013580553985950696104, −9.413442869167588406822034333128, −8.64994307223101551731605732889, −7.627812731896061139335130973642, −6.94325493376608177605107255637, −5.84991902194875672384012735650, −4.230745983328097486261142542229, −3.59400125048563062680834886119, −2.34465881061300123609852210159, −1.257402745718686308965750592187, 0.18395037104452178084614403491, 2.0430389151494575289811355424, 2.57417524859053701606711584368, 4.463317050395295388143542793426, 5.35843189514603451490721400338, 6.29806904765350963081728291687, 7.21218676569625999724739351139, 8.003195431952804214979158988335, 9.11603886380615157296356835768, 9.56606568185475100308551296133, 10.54348111010646729154916444486, 11.595646094878680463674909418666, 12.35890035976335273263862215811, 13.46856522377336172261638968942, 14.655028782014617583876988652997, 15.35311891164860152765795017675, 15.75956104241353235077744911921, 16.97862686394468658813192920164, 17.69218564173921487896751706152, 18.258614018366458972414229644678, 19.20336128543622657009970160275, 19.96511035921903674962184756667, 20.59176732922214304077380915000, 21.95254825489419645859743753974, 22.49377096746491883919048147190

Graph of the ZZ-function along the critical line