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Properties

Label 2-819-7.2-c1-0-6
Degree 22
Conductor 819819
Sign −0.811+0.583i-0.811 + 0.583i
Analytic cond. 6.539746.53974
Root an. cond. 2.557292.55729
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.08 + 1.87i)2-s + (−1.34 − 2.33i)4-s + (−0.758 + 1.31i)5-s + (1.40 + 2.24i)7-s + 1.51·8-s + (−1.64 − 2.84i)10-s + (2.04 + 3.54i)11-s − 13-s + (−5.73 + 0.208i)14-s + (1.05 − 1.82i)16-s + (2.48 + 4.29i)17-s + (1.19 − 2.07i)19-s + 4.09·20-s − 8.87·22-s + (−1.28 + 2.23i)23-s + ⋯
L(s)  = 1  + (−0.766 + 1.32i)2-s + (−0.674 − 1.16i)4-s + (−0.339 + 0.587i)5-s + (0.531 + 0.847i)7-s + 0.536·8-s + (−0.519 − 0.900i)10-s + (0.617 + 1.06i)11-s − 0.277·13-s + (−1.53 + 0.0558i)14-s + (0.263 − 0.457i)16-s + (0.601 + 1.04i)17-s + (0.275 − 0.476i)19-s + 0.915·20-s − 1.89·22-s + (−0.268 + 0.465i)23-s + ⋯

Functional equation

Λ(s)=(819s/2 ΓC(s) L(s)=((−0.811+0.583i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(819s/2 ΓC(s+1/2) L(s)=((−0.811+0.583i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 819819    =    32⋅7⋅133^{2} \cdot 7 \cdot 13
Sign: −0.811+0.583i-0.811 + 0.583i
Analytic conductor: 6.539746.53974
Root analytic conductor: 2.557292.55729
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ819(352,⋅)\chi_{819} (352, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 819, ( :1/2), −0.811+0.583i)(2,\ 819,\ (\ :1/2),\ -0.811 + 0.583i)

Particular Values

L(1)L(1) ≈\approx 0.242362−0.752203i0.242362 - 0.752203i
L(12)L(\frac12) ≈\approx 0.242362−0.752203i0.242362 - 0.752203i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
7 1+(−1.40−2.24i)T 1 + (-1.40 - 2.24i)T
13 1+T 1 + T
good2 1+(1.08−1.87i)T+(−1−1.73i)T2 1 + (1.08 - 1.87i)T + (-1 - 1.73i)T^{2}
5 1+(0.758−1.31i)T+(−2.5−4.33i)T2 1 + (0.758 - 1.31i)T + (-2.5 - 4.33i)T^{2}
11 1+(−2.04−3.54i)T+(−5.5+9.52i)T2 1 + (-2.04 - 3.54i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.48−4.29i)T+(−8.5+14.7i)T2 1 + (-2.48 - 4.29i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.19+2.07i)T+(−9.5−16.4i)T2 1 + (-1.19 + 2.07i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.28−2.23i)T+(−11.5−19.9i)T2 1 + (1.28 - 2.23i)T + (-11.5 - 19.9i)T^{2}
29 1+5.68T+29T2 1 + 5.68T + 29T^{2}
31 1+(−0.182−0.316i)T+(−15.5+26.8i)T2 1 + (-0.182 - 0.316i)T + (-15.5 + 26.8i)T^{2}
37 1+(−1.96+3.39i)T+(−18.5−32.0i)T2 1 + (-1.96 + 3.39i)T + (-18.5 - 32.0i)T^{2}
41 1+1.34T+41T2 1 + 1.34T + 41T^{2}
43 1+8.47T+43T2 1 + 8.47T + 43T^{2}
47 1+(−2.16+3.75i)T+(−23.5−40.7i)T2 1 + (-2.16 + 3.75i)T + (-23.5 - 40.7i)T^{2}
53 1+(5.08+8.79i)T+(−26.5+45.8i)T2 1 + (5.08 + 8.79i)T + (-26.5 + 45.8i)T^{2}
59 1+(5.50+9.53i)T+(−29.5+51.0i)T2 1 + (5.50 + 9.53i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.83−11.8i)T+(−30.5−52.8i)T2 1 + (6.83 - 11.8i)T + (-30.5 - 52.8i)T^{2}
67 1+(−5.44−9.42i)T+(−33.5+58.0i)T2 1 + (-5.44 - 9.42i)T + (-33.5 + 58.0i)T^{2}
71 1+6.18T+71T2 1 + 6.18T + 71T^{2}
73 1+(2.58+4.47i)T+(−36.5+63.2i)T2 1 + (2.58 + 4.47i)T + (-36.5 + 63.2i)T^{2}
79 1+(−6.47+11.2i)T+(−39.5−68.4i)T2 1 + (-6.47 + 11.2i)T + (-39.5 - 68.4i)T^{2}
83 1−11.7T+83T2 1 - 11.7T + 83T^{2}
89 1+(−1.87+3.25i)T+(−44.5−77.0i)T2 1 + (-1.87 + 3.25i)T + (-44.5 - 77.0i)T^{2}
97 1−17.2T+97T2 1 - 17.2T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.45484237743963118672867437381, −9.542841780667818587179941041785, −8.961101805039957791268109692954, −7.983154238727736162014244855274, −7.41686936029645408271463845273, −6.63157149228354777281799745510, −5.73568517054149149519000986377, −4.85929904131032808346299642667, −3.42832496978347353650243201080, −1.81723509901332654573551577768, 0.54116257578783712287751903744, 1.49056538207339677389803185395, 3.02003423736657260186756874123, 3.90646597278875259110835963972, 4.92954147562423869787209198217, 6.25455277370097968501399159156, 7.61746907557387241084124659996, 8.234173699034294064787847631094, 9.092084607678206712749825660903, 9.783603010237384320619974203234

Graph of the ZZ-function along the critical line