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Properties

Label 1-2243-2243.9-r0-0-0
Degree $1$
Conductor $2243$
Sign $0.997 - 0.0694i$
Analytic cond. $10.4164$
Root an. cond. $10.4164$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(1\)
Conductor: \(2243\)
Sign: $0.997 - 0.0694i$
Analytic conductor: \(10.4164\)
Root analytic conductor: \(10.4164\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2243} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2243,\ (0:\ ),\ 0.997 - 0.0694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.611083621 - 0.09072780883i\)
\(L(\frac12)\) \(\approx\) \(2.611083621 - 0.09072780883i\)
\(L(1)\) \(\approx\) \(1.673958891 + 0.4134045551i\)
\(L(1)\) \(\approx\) \(1.673958891 + 0.4134045551i\)

Euler product

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

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 $Z$-function along the critical line