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Properties

Label 2-431-431.101-c2-0-15
Degree 22
Conductor 431431
Sign −0.618−0.786i-0.618 - 0.786i
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  + (0.785 − 1.52i)2-s + (0.628 + 5.71i)3-s + (0.608 + 0.850i)4-s + (−4.29 − 2.01i)5-s + (9.22 + 3.52i)6-s + (2.32 + 10.4i)7-s + (8.58 − 1.26i)8-s + (−23.4 + 5.21i)9-s + (−6.45 + 4.98i)10-s + (14.2 − 7.97i)11-s + (−4.47 + 4.01i)12-s + (−14.2 + 7.34i)13-s + (17.7 + 4.64i)14-s + (8.80 − 25.8i)15-s + (3.46 − 10.1i)16-s + (7.96 + 14.1i)17-s + ⋯
L(s)  = 1  + (0.392 − 0.764i)2-s + (0.209 + 1.90i)3-s + (0.152 + 0.212i)4-s + (−0.859 − 0.402i)5-s + (1.53 + 0.587i)6-s + (0.331 + 1.48i)7-s + (1.07 − 0.157i)8-s + (−2.60 + 0.579i)9-s + (−0.645 + 0.498i)10-s + (1.29 − 0.724i)11-s + (−0.373 + 0.334i)12-s + (−1.09 + 0.564i)13-s + (1.26 + 0.331i)14-s + (0.586 − 1.72i)15-s + (0.216 − 0.634i)16-s + (0.468 + 0.834i)17-s + ⋯

Functional equation

Λ(s)=(431s/2 ΓC(s) L(s)=((−0.618−0.786i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 - 0.786i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(431s/2 ΓC(s+1) L(s)=((−0.618−0.786i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.618 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 431431
Sign: −0.618−0.786i-0.618 - 0.786i
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), −0.618−0.786i)(2,\ 431,\ (\ :1),\ -0.618 - 0.786i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.850555+1.75111i0.850555 + 1.75111i
L(12)L(\frac12) ≈\approx 0.850555+1.75111i0.850555 + 1.75111i
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+(−305.+303.i)T 1 + (-305. + 303. i)T
good2 1+(−0.785+1.52i)T+(−2.32−3.25i)T2 1 + (-0.785 + 1.52i)T + (-2.32 - 3.25i)T^{2}
3 1+(−0.628−5.71i)T+(−8.78+1.95i)T2 1 + (-0.628 - 5.71i)T + (-8.78 + 1.95i)T^{2}
5 1+(4.29+2.01i)T+(15.9+19.2i)T2 1 + (4.29 + 2.01i)T + (15.9 + 19.2i)T^{2}
7 1+(−2.32−10.4i)T+(−44.3+20.7i)T2 1 + (-2.32 - 10.4i)T + (-44.3 + 20.7i)T^{2}
11 1+(−14.2+7.97i)T+(63.0−103.i)T2 1 + (-14.2 + 7.97i)T + (63.0 - 103. i)T^{2}
13 1+(14.2−7.34i)T+(98.3−137.i)T2 1 + (14.2 - 7.34i)T + (98.3 - 137. i)T^{2}
17 1+(−7.96−14.1i)T+(−150.+246.i)T2 1 + (-7.96 - 14.1i)T + (-150. + 246. i)T^{2}
19 1+(5.01−2.35i)T+(230.−277.i)T2 1 + (5.01 - 2.35i)T + (230. - 277. i)T^{2}
23 1+(19.8+20.5i)T+(−19.3+528.i)T2 1 + (19.8 + 20.5i)T + (-19.3 + 528. i)T^{2}
29 1+(−20.7+29.0i)T+(−271.−795.i)T2 1 + (-20.7 + 29.0i)T + (-271. - 795. i)T^{2}
31 1+(−16.4−42.9i)T+(−715.+641.i)T2 1 + (-16.4 - 42.9i)T + (-715. + 641. i)T^{2}
37 1+(20.1+5.27i)T+(1.19e3+670.i)T2 1 + (20.1 + 5.27i)T + (1.19e3 + 670. i)T^{2}
41 1+(7.18−21.0i)T+(−1.33e3−1.02e3i)T2 1 + (7.18 - 21.0i)T + (-1.33e3 - 1.02e3i)T^{2}
43 1+(12.1−15.7i)T+(−467.−1.78e3i)T2 1 + (12.1 - 15.7i)T + (-467. - 1.78e3i)T^{2}
47 1+(−0.223+3.05i)T+(−2.18e3−321.i)T2 1 + (-0.223 + 3.05i)T + (-2.18e3 - 321. i)T^{2}
53 1+(40.0−2.93i)T+(2.77e3−408.i)T2 1 + (40.0 - 2.93i)T + (2.77e3 - 408. i)T^{2}
59 1+(−9.62+18.7i)T+(−2.02e3−2.83e3i)T2 1 + (-9.62 + 18.7i)T + (-2.02e3 - 2.83e3i)T^{2}
61 1+(−17.1−40.3i)T+(−2.58e3+2.67e3i)T2 1 + (-17.1 - 40.3i)T + (-2.58e3 + 2.67e3i)T^{2}
67 1+(−60.2−15.7i)T+(3.91e3+2.19e3i)T2 1 + (-60.2 - 15.7i)T + (3.91e3 + 2.19e3i)T^{2}
71 1+(7.23−98.8i)T+(−4.98e3−733.i)T2 1 + (7.23 - 98.8i)T + (-4.98e3 - 733. i)T^{2}
73 1+(−77.8−117.i)T+(−2.08e3+4.90e3i)T2 1 + (-77.8 - 117. i)T + (-2.08e3 + 4.90e3i)T^{2}
79 1+(−36.5+121.i)T+(−5.20e3−3.44e3i)T2 1 + (-36.5 + 121. i)T + (-5.20e3 - 3.44e3i)T^{2}
83 1+(−92.4+10.1i)T+(6.72e3−1.49e3i)T2 1 + (-92.4 + 10.1i)T + (6.72e3 - 1.49e3i)T^{2}
89 1+(−15.5+2.87i)T+(7.39e3−2.82e3i)T2 1 + (-15.5 + 2.87i)T + (7.39e3 - 2.82e3i)T^{2}
97 1+(−26.8+8.06i)T+(7.84e3−5.19e3i)T2 1 + (-26.8 + 8.06i)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

−11.54397383249473174173433762951, −10.42722317632223872154821295067, −9.627070894855366930976685924470, −8.509670449421917815007677417458, −8.301907944941630207511340169227, −6.22251723693815830226562850509, −4.92929368272519272741751983906, −4.25571461307045459816287542195, −3.46427355478191742887819667972, −2.35796067804111850142213292995, 0.70769052850832708673338709346, 1.90599312776580665235881758950, 3.62342601858841156724008534603, 4.98587213247194708137820393995, 6.40709215500196779545405632156, 7.04939191271752095815745208959, 7.53553903987568327510357043323, 7.963136801027684500198073315805, 9.688717257799471822500991115738, 10.92613415477423679505918488129

Graph of the ZZ-function along the critical line