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Properties

Label 2-431-431.367-c2-0-36
Degree 22
Conductor 431431
Sign 0.597−0.802i0.597 - 0.802i
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  + (0.655 + 1.27i)2-s + (−0.309 + 2.81i)3-s + (1.13 − 1.57i)4-s + (−3.99 + 1.87i)5-s + (−3.79 + 1.45i)6-s + (2.49 − 11.2i)7-s + (8.43 + 1.24i)8-s + (0.966 + 0.215i)9-s + (−5.01 − 3.87i)10-s + (−7.78 − 4.36i)11-s + (4.09 + 3.66i)12-s + (18.9 + 9.72i)13-s + (15.9 − 4.16i)14-s + (−4.03 − 11.8i)15-s + (1.43 + 4.21i)16-s + (4.96 − 8.84i)17-s + ⋯
L(s)  = 1  + (0.327 + 0.637i)2-s + (−0.103 + 0.937i)3-s + (0.282 − 0.394i)4-s + (−0.799 + 0.374i)5-s + (−0.631 + 0.241i)6-s + (0.356 − 1.60i)7-s + (1.05 + 0.155i)8-s + (0.107 + 0.0239i)9-s + (−0.501 − 0.387i)10-s + (−0.707 − 0.397i)11-s + (0.341 + 0.305i)12-s + (1.45 + 0.748i)13-s + (1.13 − 0.297i)14-s + (−0.269 − 0.788i)15-s + (0.0899 + 0.263i)16-s + (0.291 − 0.520i)17-s + ⋯

Functional equation

Λ(s)=(431s/2 ΓC(s) L(s)=((0.597−0.802i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(431s/2 ΓC(s+1) L(s)=((0.597−0.802i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 431431
Sign: 0.597−0.802i0.597 - 0.802i
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), 0.597−0.802i)(2,\ 431,\ (\ :1),\ 0.597 - 0.802i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.00982+1.00925i2.00982 + 1.00925i
L(12)L(\frac12) ≈\approx 2.00982+1.00925i2.00982 + 1.00925i
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+(313.+295.i)T 1 + (313. + 295. i)T
good2 1+(−0.655−1.27i)T+(−2.32+3.25i)T2 1 + (-0.655 - 1.27i)T + (-2.32 + 3.25i)T^{2}
3 1+(0.309−2.81i)T+(−8.78−1.95i)T2 1 + (0.309 - 2.81i)T + (-8.78 - 1.95i)T^{2}
5 1+(3.99−1.87i)T+(15.9−19.2i)T2 1 + (3.99 - 1.87i)T + (15.9 - 19.2i)T^{2}
7 1+(−2.49+11.2i)T+(−44.3−20.7i)T2 1 + (-2.49 + 11.2i)T + (-44.3 - 20.7i)T^{2}
11 1+(7.78+4.36i)T+(63.0+103.i)T2 1 + (7.78 + 4.36i)T + (63.0 + 103. i)T^{2}
13 1+(−18.9−9.72i)T+(98.3+137.i)T2 1 + (-18.9 - 9.72i)T + (98.3 + 137. i)T^{2}
17 1+(−4.96+8.84i)T+(−150.−246.i)T2 1 + (-4.96 + 8.84i)T + (-150. - 246. i)T^{2}
19 1+(−23.6−11.0i)T+(230.+277.i)T2 1 + (-23.6 - 11.0i)T + (230. + 277. i)T^{2}
23 1+(0.0572−0.0593i)T+(−19.3−528.i)T2 1 + (0.0572 - 0.0593i)T + (-19.3 - 528. i)T^{2}
29 1+(3.01+4.22i)T+(−271.+795.i)T2 1 + (3.01 + 4.22i)T + (-271. + 795. i)T^{2}
31 1+(−8.48+22.1i)T+(−715.−641.i)T2 1 + (-8.48 + 22.1i)T + (-715. - 641. i)T^{2}
37 1+(−16.0+4.20i)T+(1.19e3−670.i)T2 1 + (-16.0 + 4.20i)T + (1.19e3 - 670. i)T^{2}
41 1+(17.0+49.9i)T+(−1.33e3+1.02e3i)T2 1 + (17.0 + 49.9i)T + (-1.33e3 + 1.02e3i)T^{2}
43 1+(−30.6−39.6i)T+(−467.+1.78e3i)T2 1 + (-30.6 - 39.6i)T + (-467. + 1.78e3i)T^{2}
47 1+(−5.84−79.8i)T+(−2.18e3+321.i)T2 1 + (-5.84 - 79.8i)T + (-2.18e3 + 321. i)T^{2}
53 1+(31.4+2.30i)T+(2.77e3+408.i)T2 1 + (31.4 + 2.30i)T + (2.77e3 + 408. i)T^{2}
59 1+(17.8+34.6i)T+(−2.02e3+2.83e3i)T2 1 + (17.8 + 34.6i)T + (-2.02e3 + 2.83e3i)T^{2}
61 1+(−25.2+59.3i)T+(−2.58e3−2.67e3i)T2 1 + (-25.2 + 59.3i)T + (-2.58e3 - 2.67e3i)T^{2}
67 1+(−41.9+10.9i)T+(3.91e3−2.19e3i)T2 1 + (-41.9 + 10.9i)T + (3.91e3 - 2.19e3i)T^{2}
71 1+(−5.78−79.0i)T+(−4.98e3+733.i)T2 1 + (-5.78 - 79.0i)T + (-4.98e3 + 733. i)T^{2}
73 1+(−32.4+49.0i)T+(−2.08e3−4.90e3i)T2 1 + (-32.4 + 49.0i)T + (-2.08e3 - 4.90e3i)T^{2}
79 1+(−0.563−1.87i)T+(−5.20e3+3.44e3i)T2 1 + (-0.563 - 1.87i)T + (-5.20e3 + 3.44e3i)T^{2}
83 1+(27.0+2.97i)T+(6.72e3+1.49e3i)T2 1 + (27.0 + 2.97i)T + (6.72e3 + 1.49e3i)T^{2}
89 1+(45.7+8.44i)T+(7.39e3+2.82e3i)T2 1 + (45.7 + 8.44i)T + (7.39e3 + 2.82e3i)T^{2}
97 1+(−175.−52.8i)T+(7.84e3+5.19e3i)T2 1 + (-175. - 52.8i)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

−11.11498716096385543396195568249, −10.35580356778599808445011139929, −9.518425888897109157191242350004, −7.80992832771081798070466132773, −7.51797505351322667416812988050, −6.38162071102281087456244906017, −5.20646517341087110923733835250, −4.21992200847619011892767436596, −3.57784774904295415000032410493, −1.15891738718577041216107493835, 1.24762802872776713330514184796, 2.46491847098501666984979577588, 3.57039396364023630224419536229, 4.95193236338569908079369303210, 6.02151366123723915826499177017, 7.29237400532503913817910122881, 8.075723180138593860047502875847, 8.629683083065234740912829289843, 10.15486812796991951007780107848, 11.30249383086948403854533194677

Graph of the ZZ-function along the critical line