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Properties

Label 1-2303-2303.281-r1-0-0
Degree 11
Conductor 23032303
Sign 0.404−0.914i0.404 - 0.914i
Analytic cond. 247.491247.491
Root an. cond. 247.491247.491
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  + (−0.900 + 0.433i)2-s + (−0.222 + 0.974i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.900 − 0.433i)11-s + (0.623 + 0.781i)12-s + (0.900 − 0.433i)13-s + (0.900 + 0.433i)15-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s + 18-s − 19-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (−0.222 + 0.974i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.900 − 0.433i)11-s + (0.623 + 0.781i)12-s + (0.900 − 0.433i)13-s + (0.900 + 0.433i)15-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s + 18-s − 19-s + ⋯

Functional equation

Λ(s)=(2303s/2 ΓR(s+1) L(s)=((0.404−0.914i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2303s/2 ΓR(s+1) L(s)=((0.404−0.914i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 23032303    =    72⋅477^{2} \cdot 47
Sign: 0.404−0.914i0.404 - 0.914i
Analytic conductor: 247.491247.491
Root analytic conductor: 247.491247.491
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2303(281,⋅)\chi_{2303} (281, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2303, (1: ), 0.404−0.914i)(1,\ 2303,\ (1:\ ),\ 0.404 - 0.914i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.7644435703−0.4975976215i0.7644435703 - 0.4975976215i
L(12)L(\frac12) ≈\approx 0.7644435703−0.4975976215i0.7644435703 - 0.4975976215i
L(1)L(1) ≈\approx 0.6829317525+0.1187070421i0.6829317525 + 0.1187070421i
L(1)L(1) ≈\approx 0.6829317525+0.1187070421i0.6829317525 + 0.1187070421i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
47 1 1
good2 1+(−0.900+0.433i)T 1 + (-0.900 + 0.433i)T
3 1+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
5 1+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
11 1+(0.900−0.433i)T 1 + (0.900 - 0.433i)T
13 1+(0.900−0.433i)T 1 + (0.900 - 0.433i)T
17 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
19 1−T 1 - T
23 1+(−0.623+0.781i)T 1 + (-0.623 + 0.781i)T
29 1+(−0.623−0.781i)T 1 + (-0.623 - 0.781i)T
31 1−T 1 - T
37 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
41 1+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
43 1+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
53 1+(0.623−0.781i)T 1 + (0.623 - 0.781i)T
59 1+(−0.222−0.974i)T 1 + (-0.222 - 0.974i)T
61 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
67 1−T 1 - T
71 1+(0.623−0.781i)T 1 + (0.623 - 0.781i)T
73 1+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
79 1+T 1 + T
83 1+(−0.900−0.433i)T 1 + (-0.900 - 0.433i)T
89 1+(−0.900−0.433i)T 1 + (-0.900 - 0.433i)T
97 1+T 1 + 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

−19.41956917180007768877040817206, −18.67679739288227938924089176428, −18.24851703555040159942420731755, −17.82134113795869258460985935794, −16.713004988631238376315301457854, −16.556527974828816074116534647635, −15.24547368252857881402960480762, −14.41999527535568375709279626207, −13.82107892047609708608327304021, −12.793754011806742488058588638688, −12.251106650714253352145752786958, −11.33685021126143290553251004250, −11.023863838013678607900488439463, −10.14183895469788263820885041333, −9.220199138134080485868550746634, −8.59112473375367060834316794981, −7.5911404295972950291883072158, −7.05520898725509500805445596172, −6.41900766441700081988598167212, −5.77463759171141363811609810392, −4.1473915554602165317675448264, −3.32420394280968062605478540910, −2.33418328692735688969678270267, −1.79238380909047950063149256215, −0.85597770290885590461184313161, 0.26976788772357063452079610691, 1.1531617760877922792784376204, 2.04746825335380868044335583763, 3.52196425576459615130175823613, 4.150621812948483780892050734873, 5.26283537531244213575758061817, 5.936501630919253288534563845564, 6.3195763786682758931014955318, 7.79787485168278877436301293086, 8.436181802738497273934902414165, 9.01008339563993561691870044193, 9.65445079896002775662091312923, 10.33947343112802512626625355926, 11.16374432418894260466999890944, 11.72072836784383891791247849777, 12.72369852128269797365522516361, 13.72249874920147713068259206423, 14.585394775716736461078441757419, 15.21550862516661855424428662481, 15.96402984028121163068671102240, 16.55831496172411595695993287517, 17.06436236350627825608045747763, 17.562730421994283249599503411968, 18.48511385755417105492606229476, 19.5417689798647478452434374298

Graph of the ZZ-function along the critical line