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Properties

Label 2-431-431.367-c2-0-69
Degree 22
Conductor 431431
Sign −0.500−0.865i-0.500 - 0.865i
Analytic cond. 11.743811.7438
Root an. cond. 3.426933.42693
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.419 − 0.816i)2-s + (0.503 − 4.57i)3-s + (1.83 − 2.56i)4-s + (−5.01 + 2.34i)5-s + (−3.94 + 1.50i)6-s + (1.85 − 8.34i)7-s + (−6.49 − 0.956i)8-s + (−11.8 − 2.64i)9-s + (4.01 + 3.10i)10-s + (−5.44 − 3.05i)11-s + (−10.8 − 9.69i)12-s + (5.77 + 2.96i)13-s + (−7.59 + 1.98i)14-s + (8.22 + 24.0i)15-s + (−2.13 − 6.24i)16-s + (−9.26 + 16.5i)17-s + ⋯
L(s)  = 1  + (−0.209 − 0.408i)2-s + (0.167 − 1.52i)3-s + (0.459 − 0.642i)4-s + (−1.00 + 0.469i)5-s + (−0.657 + 0.251i)6-s + (0.265 − 1.19i)7-s + (−0.812 − 0.119i)8-s + (−1.32 − 0.294i)9-s + (0.401 + 0.310i)10-s + (−0.494 − 0.277i)11-s + (−0.901 − 0.807i)12-s + (0.444 + 0.228i)13-s + (−0.542 + 0.141i)14-s + (0.548 + 1.60i)15-s + (−0.133 − 0.390i)16-s + (−0.545 + 0.971i)17-s + ⋯

Functional equation

Λ(s)=(431s/2 ΓC(s) L(s)=((−0.500−0.865i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.865i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(431s/2 ΓC(s+1) L(s)=((−0.500−0.865i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 431431
Sign: −0.500−0.865i-0.500 - 0.865i
Analytic conductor: 11.743811.7438
Root analytic conductor: 3.426933.42693
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ431(367,⋅)\chi_{431} (367, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 431, ( :1), −0.500−0.865i)(2,\ 431,\ (\ :1),\ -0.500 - 0.865i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.497299+0.861686i0.497299 + 0.861686i
L(12)L(\frac12) ≈\approx 0.497299+0.861686i0.497299 + 0.861686i
L(2)L(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)
bad431 1+(395.−171.i)T 1 + (395. - 171. i)T
good2 1+(0.419+0.816i)T+(−2.32+3.25i)T2 1 + (0.419 + 0.816i)T + (-2.32 + 3.25i)T^{2}
3 1+(−0.503+4.57i)T+(−8.78−1.95i)T2 1 + (-0.503 + 4.57i)T + (-8.78 - 1.95i)T^{2}
5 1+(5.01−2.34i)T+(15.9−19.2i)T2 1 + (5.01 - 2.34i)T + (15.9 - 19.2i)T^{2}
7 1+(−1.85+8.34i)T+(−44.3−20.7i)T2 1 + (-1.85 + 8.34i)T + (-44.3 - 20.7i)T^{2}
11 1+(5.44+3.05i)T+(63.0+103.i)T2 1 + (5.44 + 3.05i)T + (63.0 + 103. i)T^{2}
13 1+(−5.77−2.96i)T+(98.3+137.i)T2 1 + (-5.77 - 2.96i)T + (98.3 + 137. i)T^{2}
17 1+(9.26−16.5i)T+(−150.−246.i)T2 1 + (9.26 - 16.5i)T + (-150. - 246. i)T^{2}
19 1+(−13.6−6.39i)T+(230.+277.i)T2 1 + (-13.6 - 6.39i)T + (230. + 277. i)T^{2}
23 1+(−14.2+14.8i)T+(−19.3−528.i)T2 1 + (-14.2 + 14.8i)T + (-19.3 - 528. i)T^{2}
29 1+(27.8+38.9i)T+(−271.+795.i)T2 1 + (27.8 + 38.9i)T + (-271. + 795. i)T^{2}
31 1+(−5.59+14.6i)T+(−715.−641.i)T2 1 + (-5.59 + 14.6i)T + (-715. - 641. i)T^{2}
37 1+(−40.1+10.4i)T+(1.19e3−670.i)T2 1 + (-40.1 + 10.4i)T + (1.19e3 - 670. i)T^{2}
41 1+(−11.4−33.6i)T+(−1.33e3+1.02e3i)T2 1 + (-11.4 - 33.6i)T + (-1.33e3 + 1.02e3i)T^{2}
43 1+(−15.3−19.9i)T+(−467.+1.78e3i)T2 1 + (-15.3 - 19.9i)T + (-467. + 1.78e3i)T^{2}
47 1+(−1.80−24.6i)T+(−2.18e3+321.i)T2 1 + (-1.80 - 24.6i)T + (-2.18e3 + 321. i)T^{2}
53 1+(54.5+3.98i)T+(2.77e3+408.i)T2 1 + (54.5 + 3.98i)T + (2.77e3 + 408. i)T^{2}
59 1+(23.8+46.3i)T+(−2.02e3+2.83e3i)T2 1 + (23.8 + 46.3i)T + (-2.02e3 + 2.83e3i)T^{2}
61 1+(17.3−40.8i)T+(−2.58e3−2.67e3i)T2 1 + (17.3 - 40.8i)T + (-2.58e3 - 2.67e3i)T^{2}
67 1+(11.5−3.02i)T+(3.91e3−2.19e3i)T2 1 + (11.5 - 3.02i)T + (3.91e3 - 2.19e3i)T^{2}
71 1+(4.96+67.7i)T+(−4.98e3+733.i)T2 1 + (4.96 + 67.7i)T + (-4.98e3 + 733. i)T^{2}
73 1+(−29.8+45.0i)T+(−2.08e3−4.90e3i)T2 1 + (-29.8 + 45.0i)T + (-2.08e3 - 4.90e3i)T^{2}
79 1+(−20.9−69.4i)T+(−5.20e3+3.44e3i)T2 1 + (-20.9 - 69.4i)T + (-5.20e3 + 3.44e3i)T^{2}
83 1+(37.5+4.13i)T+(6.72e3+1.49e3i)T2 1 + (37.5 + 4.13i)T + (6.72e3 + 1.49e3i)T^{2}
89 1+(−57.9−10.6i)T+(7.39e3+2.82e3i)T2 1 + (-57.9 - 10.6i)T + (7.39e3 + 2.82e3i)T^{2}
97 1+(44.2+13.3i)T+(7.84e3+5.19e3i)T2 1 + (44.2 + 13.3i)T + (7.84e3 + 5.19e3i)T^{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

−10.85039614011264403238531078829, −9.521420017147103690302323657144, −8.014710838496338964034065098696, −7.67901382870521426884651558746, −6.71757346241054304041074598313, −5.99660512521659565195538918586, −4.14286249312453269678193159210, −2.83210303467570236739941426198, −1.52139626717477113027983122977, −0.44245067079298002892072625836, 2.73857441789454484382047152202, 3.63249306277537770288436100763, 4.81858497440193107859139622322, 5.55985753653647814566614383373, 7.17041203278742969471126459191, 8.075914199964639838915809882631, 8.961095280024505664791628560485, 9.299360450113093874379410834869, 10.79457528164002130161054911614, 11.46630354294182938352027668414

Graph of the ZZ-function along the critical line