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Properties

Label 2-431-431.428-c2-0-40
Degree 22
Conductor 431431
Sign 0.986−0.161i0.986 - 0.161i
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  + (0.468 + 0.485i)2-s + (1.59 + 2.61i)3-s + (0.129 − 3.54i)4-s + (1.20 − 1.68i)5-s + (−0.523 + 2.00i)6-s + (8.95 + 4.60i)7-s + (3.78 − 3.39i)8-s + (−0.190 + 0.369i)9-s + (1.37 − 0.203i)10-s + (−6.35 − 4.20i)11-s + (9.48 − 5.32i)12-s + (4.93 + 4.75i)13-s + (1.95 + 6.49i)14-s + (6.32 + 0.463i)15-s + (−10.7 − 0.784i)16-s + (9.97 − 15.0i)17-s + ⋯
L(s)  = 1  + (0.234 + 0.242i)2-s + (0.532 + 0.873i)3-s + (0.0323 − 0.885i)4-s + (0.240 − 0.336i)5-s + (−0.0872 + 0.333i)6-s + (1.27 + 0.657i)7-s + (0.473 − 0.424i)8-s + (−0.0211 + 0.0411i)9-s + (0.137 − 0.0203i)10-s + (−0.577 − 0.382i)11-s + (0.790 − 0.443i)12-s + (0.379 + 0.366i)13-s + (0.139 + 0.464i)14-s + (0.421 + 0.0308i)15-s + (−0.670 − 0.0490i)16-s + (0.586 − 0.886i)17-s + ⋯

Functional equation

Λ(s)=(431s/2 ΓC(s) L(s)=((0.986−0.161i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(431s/2 ΓC(s+1) L(s)=((0.986−0.161i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 431431
Sign: 0.986−0.161i0.986 - 0.161i
Analytic conductor: 11.743811.7438
Root analytic conductor: 3.426933.42693
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ431(428,⋅)\chi_{431} (428, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 431, ( :1), 0.986−0.161i)(2,\ 431,\ (\ :1),\ 0.986 - 0.161i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.82300+0.229247i2.82300 + 0.229247i
L(12)L(\frac12) ≈\approx 2.82300+0.229247i2.82300 + 0.229247i
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+(134.−409.i)T 1 + (134. - 409. i)T
good2 1+(−0.468−0.485i)T+(−0.146+3.99i)T2 1 + (-0.468 - 0.485i)T + (-0.146 + 3.99i)T^{2}
3 1+(−1.59−2.61i)T+(−4.11+8.00i)T2 1 + (-1.59 - 2.61i)T + (-4.11 + 8.00i)T^{2}
5 1+(−1.20+1.68i)T+(−8.07−23.6i)T2 1 + (-1.20 + 1.68i)T + (-8.07 - 23.6i)T^{2}
7 1+(−8.95−4.60i)T+(28.5+39.8i)T2 1 + (-8.95 - 4.60i)T + (28.5 + 39.8i)T^{2}
11 1+(6.35+4.20i)T+(47.3+111.i)T2 1 + (6.35 + 4.20i)T + (47.3 + 111. i)T^{2}
13 1+(−4.93−4.75i)T+(6.17+168.i)T2 1 + (-4.93 - 4.75i)T + (6.17 + 168. i)T^{2}
17 1+(−9.97+15.0i)T+(−113.−265.i)T2 1 + (-9.97 + 15.0i)T + (-113. - 265. i)T^{2}
19 1+(14.2+19.8i)T+(−116.+341.i)T2 1 + (14.2 + 19.8i)T + (-116. + 341. i)T^{2}
23 1+(−9.67−11.6i)T+(−96.0+520.i)T2 1 + (-9.67 - 11.6i)T + (-96.0 + 520. i)T^{2}
29 1+(0.100+2.74i)T+(−838.+61.3i)T2 1 + (0.100 + 2.74i)T + (-838. + 61.3i)T^{2}
31 1+(−5.22+1.36i)T+(838.−470.i)T2 1 + (-5.22 + 1.36i)T + (838. - 470. i)T^{2}
37 1+(−4.46−14.8i)T+(−1.14e3+755.i)T2 1 + (-4.46 - 14.8i)T + (-1.14e3 + 755. i)T^{2}
41 1+(−32.6−2.38i)T+(1.66e3+244.i)T2 1 + (-32.6 - 2.38i)T + (1.66e3 + 244. i)T^{2}
43 1+(7.83−53.2i)T+(−1.77e3−532.i)T2 1 + (7.83 - 53.2i)T + (-1.77e3 - 532. i)T^{2}
47 1+(−8.23+21.5i)T+(−1.64e3−1.47e3i)T2 1 + (-8.23 + 21.5i)T + (-1.64e3 - 1.47e3i)T^{2}
53 1+(47.7−18.2i)T+(2.09e3−1.87e3i)T2 1 + (47.7 - 18.2i)T + (2.09e3 - 1.87e3i)T^{2}
59 1+(−20.3−21.1i)T+(−127.+3.47e3i)T2 1 + (-20.3 - 21.1i)T + (-127. + 3.47e3i)T^{2}
61 1+(60.7−28.4i)T+(2.38e3−2.86e3i)T2 1 + (60.7 - 28.4i)T + (2.38e3 - 2.86e3i)T^{2}
67 1+(−4.09−13.6i)T+(−3.74e3+2.47e3i)T2 1 + (-4.09 - 13.6i)T + (-3.74e3 + 2.47e3i)T^{2}
71 1+(23.9−62.6i)T+(−3.75e3−3.36e3i)T2 1 + (23.9 - 62.6i)T + (-3.75e3 - 3.36e3i)T^{2}
73 1+(3.79−17.0i)T+(−4.82e3−2.26e3i)T2 1 + (3.79 - 17.0i)T + (-4.82e3 - 2.26e3i)T^{2}
79 1+(113.−12.5i)T+(6.09e3−1.35e3i)T2 1 + (113. - 12.5i)T + (6.09e3 - 1.35e3i)T^{2}
83 1+(8.34−5.09i)T+(3.14e3−6.12e3i)T2 1 + (8.34 - 5.09i)T + (3.14e3 - 6.12e3i)T^{2}
89 1+(−42.4+54.9i)T+(−2.00e3−7.66e3i)T2 1 + (-42.4 + 54.9i)T + (-2.00e3 - 7.66e3i)T^{2}
97 1+(−17.7+161.i)T+(−9.18e3−2.04e3i)T2 1 + (-17.7 + 161. i)T + (-9.18e3 - 2.04e3i)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

−10.99564979566923912473378329689, −9.927067122220901717454135921620, −9.157118234635019351429834242493, −8.537716035022160365458160245904, −7.26657261845087847133644149841, −5.92519063750801597950612984052, −5.02756596342150210639032490781, −4.46804124654510730770525369076, −2.79761797217816188597159997668, −1.29529483612786813411561403737, 1.57860583774240031244119104098, 2.51306077297812044347728201857, 3.86445091604725586348798161434, 4.90798676346583394803043143658, 6.43605110345122437718080741280, 7.64210210717608029592189704966, 7.902165340098530654130890873760, 8.655368364097344414707468200866, 10.52047648429413114470863180068, 10.72900049844882353721529739571

Graph of the ZZ-function along the critical line