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Properties

Label 2-431-431.101-c2-0-53
Degree 22
Conductor 431431
Sign 0.203+0.979i0.203 + 0.979i
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  + (−1.00 + 1.95i)2-s + (−0.109 − 0.998i)3-s + (−0.488 − 0.683i)4-s + (0.139 + 0.0654i)5-s + (2.06 + 0.789i)6-s + (−0.973 − 4.36i)7-s + (−6.87 + 1.01i)8-s + (7.80 − 1.73i)9-s + (−0.268 + 0.207i)10-s + (0.328 − 0.184i)11-s + (−0.628 + 0.563i)12-s + (−15.4 + 7.94i)13-s + (9.52 + 2.49i)14-s + (0.0499 − 0.146i)15-s + (6.02 − 17.6i)16-s + (−10.7 − 19.2i)17-s + ⋯
L(s)  = 1  + (−0.502 + 0.978i)2-s + (−0.0366 − 0.332i)3-s + (−0.122 − 0.170i)4-s + (0.0279 + 0.0130i)5-s + (0.343 + 0.131i)6-s + (−0.139 − 0.624i)7-s + (−0.859 + 0.126i)8-s + (0.866 − 0.193i)9-s + (−0.0268 + 0.0207i)10-s + (0.0298 − 0.0167i)11-s + (−0.0523 + 0.0469i)12-s + (−1.18 + 0.610i)13-s + (0.680 + 0.177i)14-s + (0.00333 − 0.00977i)15-s + (0.376 − 1.10i)16-s + (−0.634 − 1.13i)17-s + ⋯

Functional equation

Λ(s)=(431s/2 ΓC(s) L(s)=((0.203+0.979i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(431s/2 ΓC(s+1) L(s)=((0.203+0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 431431
Sign: 0.203+0.979i0.203 + 0.979i
Analytic conductor: 11.743811.7438
Root analytic conductor: 3.426933.42693
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ431(101,⋅)\chi_{431} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 431, ( :1), 0.203+0.979i)(2,\ 431,\ (\ :1),\ 0.203 + 0.979i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.395452−0.321616i0.395452 - 0.321616i
L(12)L(\frac12) ≈\approx 0.395452−0.321616i0.395452 - 0.321616i
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+(409.−135.i)T 1 + (409. - 135. i)T
good2 1+(1.00−1.95i)T+(−2.32−3.25i)T2 1 + (1.00 - 1.95i)T + (-2.32 - 3.25i)T^{2}
3 1+(0.109+0.998i)T+(−8.78+1.95i)T2 1 + (0.109 + 0.998i)T + (-8.78 + 1.95i)T^{2}
5 1+(−0.139−0.0654i)T+(15.9+19.2i)T2 1 + (-0.139 - 0.0654i)T + (15.9 + 19.2i)T^{2}
7 1+(0.973+4.36i)T+(−44.3+20.7i)T2 1 + (0.973 + 4.36i)T + (-44.3 + 20.7i)T^{2}
11 1+(−0.328+0.184i)T+(63.0−103.i)T2 1 + (-0.328 + 0.184i)T + (63.0 - 103. i)T^{2}
13 1+(15.4−7.94i)T+(98.3−137.i)T2 1 + (15.4 - 7.94i)T + (98.3 - 137. i)T^{2}
17 1+(10.7+19.2i)T+(−150.+246.i)T2 1 + (10.7 + 19.2i)T + (-150. + 246. i)T^{2}
19 1+(6.11−2.86i)T+(230.−277.i)T2 1 + (6.11 - 2.86i)T + (230. - 277. i)T^{2}
23 1+(−4.13−4.28i)T+(−19.3+528.i)T2 1 + (-4.13 - 4.28i)T + (-19.3 + 528. i)T^{2}
29 1+(15.7−22.0i)T+(−271.−795.i)T2 1 + (15.7 - 22.0i)T + (-271. - 795. i)T^{2}
31 1+(4.38+11.4i)T+(−715.+641.i)T2 1 + (4.38 + 11.4i)T + (-715. + 641. i)T^{2}
37 1+(42.7+11.1i)T+(1.19e3+670.i)T2 1 + (42.7 + 11.1i)T + (1.19e3 + 670. i)T^{2}
41 1+(−20.0+58.7i)T+(−1.33e3−1.02e3i)T2 1 + (-20.0 + 58.7i)T + (-1.33e3 - 1.02e3i)T^{2}
43 1+(4.87−6.31i)T+(−467.−1.78e3i)T2 1 + (4.87 - 6.31i)T + (-467. - 1.78e3i)T^{2}
47 1+(−2.11+28.8i)T+(−2.18e3−321.i)T2 1 + (-2.11 + 28.8i)T + (-2.18e3 - 321. i)T^{2}
53 1+(−23.4+1.71i)T+(2.77e3−408.i)T2 1 + (-23.4 + 1.71i)T + (2.77e3 - 408. i)T^{2}
59 1+(3.51−6.84i)T+(−2.02e3−2.83e3i)T2 1 + (3.51 - 6.84i)T + (-2.02e3 - 2.83e3i)T^{2}
61 1+(6.56+15.4i)T+(−2.58e3+2.67e3i)T2 1 + (6.56 + 15.4i)T + (-2.58e3 + 2.67e3i)T^{2}
67 1+(66.9+17.5i)T+(3.91e3+2.19e3i)T2 1 + (66.9 + 17.5i)T + (3.91e3 + 2.19e3i)T^{2}
71 1+(−7.50+102.i)T+(−4.98e3−733.i)T2 1 + (-7.50 + 102. i)T + (-4.98e3 - 733. i)T^{2}
73 1+(−22.8−34.4i)T+(−2.08e3+4.90e3i)T2 1 + (-22.8 - 34.4i)T + (-2.08e3 + 4.90e3i)T^{2}
79 1+(19.2−64.1i)T+(−5.20e3−3.44e3i)T2 1 + (19.2 - 64.1i)T + (-5.20e3 - 3.44e3i)T^{2}
83 1+(−5.65+0.622i)T+(6.72e3−1.49e3i)T2 1 + (-5.65 + 0.622i)T + (6.72e3 - 1.49e3i)T^{2}
89 1+(−127.+23.5i)T+(7.39e3−2.82e3i)T2 1 + (-127. + 23.5i)T + (7.39e3 - 2.82e3i)T^{2}
97 1+(−50.0+15.0i)T+(7.84e3−5.19e3i)T2 1 + (-50.0 + 15.0i)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

−10.53839809753073260860397683665, −9.574397163915111058335994954869, −8.901273086083343443372352018500, −7.56732057791266656991309270303, −7.17973739172435972433422834320, −6.48881891905290148071100692047, −5.14071591715299977418911914067, −3.89989794271644133914870236308, −2.24253153020675986458394053562, −0.24185149160479236502784151272, 1.65586474444848105417588684054, 2.71669069897598066124082091566, 4.04448012271663864374975054519, 5.31613112521306900405238206939, 6.40157838064940685907001945193, 7.62650783676322650776697101764, 8.799490099658081561809475842668, 9.586019737470834195050696386761, 10.25849551689229628731326429497, 10.92873232383269275278457047999

Graph of the ZZ-function along the critical line