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Properties

Label 1-2243-2243.9-r0-0-0
Degree 11
Conductor 22432243
Sign 0.997−0.0694i0.997 - 0.0694i
Analytic cond. 10.416410.4164
Root an. cond. 10.416410.4164
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  + (0.683 + 0.730i)2-s + (0.995 + 0.0951i)3-s + (−0.0658 + 0.997i)4-s + (−0.993 − 0.114i)5-s + (0.610 + 0.791i)6-s + (−0.215 − 0.976i)7-s + (−0.773 + 0.633i)8-s + (0.981 + 0.189i)9-s + (−0.595 − 0.803i)10-s + (0.691 + 0.722i)11-s + (−0.160 + 0.987i)12-s + (0.806 − 0.591i)13-s + (0.565 − 0.824i)14-s + (−0.977 − 0.208i)15-s + (−0.991 − 0.131i)16-s + (−0.823 − 0.566i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.730i)2-s + (0.995 + 0.0951i)3-s + (−0.0658 + 0.997i)4-s + (−0.993 − 0.114i)5-s + (0.610 + 0.791i)6-s + (−0.215 − 0.976i)7-s + (−0.773 + 0.633i)8-s + (0.981 + 0.189i)9-s + (−0.595 − 0.803i)10-s + (0.691 + 0.722i)11-s + (−0.160 + 0.987i)12-s + (0.806 − 0.591i)13-s + (0.565 − 0.824i)14-s + (−0.977 − 0.208i)15-s + (−0.991 − 0.131i)16-s + (−0.823 − 0.566i)17-s + ⋯

Functional equation

Λ(s)=(2243s/2 ΓR(s) L(s)=((0.997−0.0694i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2243s/2 ΓR(s) L(s)=((0.997−0.0694i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 22432243
Sign: 0.997−0.0694i0.997 - 0.0694i
Analytic conductor: 10.416410.4164
Root analytic conductor: 10.416410.4164
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2243(9,⋅)\chi_{2243} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2243, (0: ), 0.997−0.0694i)(1,\ 2243,\ (0:\ ),\ 0.997 - 0.0694i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.611083621−0.09072780883i2.611083621 - 0.09072780883i
L(12)L(\frac12) ≈\approx 2.611083621−0.09072780883i2.611083621 - 0.09072780883i
L(1)L(1) ≈\approx 1.673958891+0.4134045551i1.673958891 + 0.4134045551i
L(1)L(1) ≈\approx 1.673958891+0.4134045551i1.673958891 + 0.4134045551i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2243 1 1
good2 1+(0.683+0.730i)T 1 + (0.683 + 0.730i)T
3 1+(0.995+0.0951i)T 1 + (0.995 + 0.0951i)T
5 1+(−0.993−0.114i)T 1 + (-0.993 - 0.114i)T
7 1+(−0.215−0.976i)T 1 + (-0.215 - 0.976i)T
11 1+(0.691+0.722i)T 1 + (0.691 + 0.722i)T
13 1+(0.806−0.591i)T 1 + (0.806 - 0.591i)T
17 1+(−0.823−0.566i)T 1 + (-0.823 - 0.566i)T
19 1+(0.583−0.811i)T 1 + (0.583 - 0.811i)T
23 1+(−0.911−0.410i)T 1 + (-0.911 - 0.410i)T
29 1+(0.336−0.941i)T 1 + (0.336 - 0.941i)T
31 1+(−0.264−0.964i)T 1 + (-0.264 - 0.964i)T
37 1+(0.146−0.989i)T 1 + (0.146 - 0.989i)T
41 1+(−0.689−0.724i)T 1 + (-0.689 - 0.724i)T
43 1+(−0.939−0.343i)T 1 + (-0.939 - 0.343i)T
47 1+(−0.652+0.758i)T 1 + (-0.652 + 0.758i)T
53 1+(−0.0937−0.995i)T 1 + (-0.0937 - 0.995i)T
59 1+(0.991+0.128i)T 1 + (0.991 + 0.128i)T
61 1+(0.898+0.438i)T 1 + (0.898 + 0.438i)T
67 1+(−0.998−0.0476i)T 1 + (-0.998 - 0.0476i)T
71 1+(0.570+0.821i)T 1 + (0.570 + 0.821i)T
73 1+(0.901+0.433i)T 1 + (0.901 + 0.433i)T
79 1+(−0.116−0.993i)T 1 + (-0.116 - 0.993i)T
83 1+(0.861+0.507i)T 1 + (0.861 + 0.507i)T
89 1+(0.711+0.702i)T 1 + (0.711 + 0.702i)T
97 1+(−0.259−0.965i)T 1 + (-0.259 - 0.965i)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

−19.744577897441914464586565068427, −19.24990584764368409753075463317, −18.45562366693295853020908962161, −18.209777300071005631723676532100, −16.32386041157489887838541196643, −15.93947774299036270113266649292, −15.12390077288019317215762195312, −14.624667321031057139953566242400, −13.86643363694426911170963135815, −13.23080055943715127452983873497, −12.30866387032902123620206541583, −11.84311038013732350569643574096, −11.153563306985265199911878469371, −10.20893939461379755851799474225, −9.3105607516359200081297072335, −8.61413803747491707373370134246, −8.17609211943650757725944079129, −6.731190914549415606441540072139, −6.34911313150642250828386888026, −5.14629115246839466857283076769, −4.19076296086415886437946782943, −3.43850853489159200365897899763, −3.17677041632425983542907116884, −1.92312014988178325539378561789, −1.29183156222535391565715562843, 0.60987475788904650682345348531, 2.142428251261067803133302771323, 3.15148716826653458221267364262, 3.98099347036183713124186453540, 4.177860767883027594633931066104, 5.11747968520624911377191434352, 6.54756709658361172558018727763, 7.03082116760457969868565485428, 7.740017187542747454731220686993, 8.31963856476961790147221812299, 9.12662349387616360850598596858, 9.95533998058260466635444347643, 11.08939516872937290360357551969, 11.7919218214187559422017943900, 12.73221133671509275438139989436, 13.3647983843831677492149898127, 13.872095936207425427237132149979, 14.74079186243545810053461151306, 15.28734487454936616306187636362, 15.994202834164534642920252066268, 16.359143344839566128797714430327, 17.51677420247901969818890090159, 18.06896116887093137843647692349, 19.18757464360162763552686696031, 19.982583998110927847130663972773

Graph of the ZZ-function along the critical line