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Properties

Label 2-832-52.7-c1-0-3
Degree 22
Conductor 832832
Sign 0.533−0.846i0.533 - 0.846i
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.36 − 1.36i)3-s + (0.633 + 0.633i)5-s + (0.267 − i)7-s + (2.23 + 3.86i)9-s + (−4.73 + 1.26i)11-s + (3.23 − 1.59i)13-s + (−0.633 − 2.36i)15-s + (−5.59 + 3.23i)17-s + (3.09 + 0.830i)19-s + (−2 + 2i)21-s + (−2.36 + 4.09i)23-s − 4.19i·25-s − 4.00i·27-s + (−1.5 + 2.59i)29-s + (3.73 − 3.73i)31-s + ⋯
L(s)  = 1  + (−1.36 − 0.788i)3-s + (0.283 + 0.283i)5-s + (0.101 − 0.377i)7-s + (0.744 + 1.28i)9-s + (−1.42 + 0.382i)11-s + (0.896 − 0.443i)13-s + (−0.163 − 0.610i)15-s + (−1.35 + 0.783i)17-s + (0.710 + 0.190i)19-s + (−0.436 + 0.436i)21-s + (−0.493 + 0.854i)23-s − 0.839i·25-s − 0.769i·27-s + (−0.278 + 0.482i)29-s + (0.670 − 0.670i)31-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((0.533−0.846i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((0.533−0.846i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 0.533−0.846i0.533 - 0.846i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(319,⋅)\chi_{832} (319, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), 0.533−0.846i)(2,\ 832,\ (\ :1/2),\ 0.533 - 0.846i)

Particular Values

L(1)L(1) ≈\approx 0.548917+0.302941i0.548917 + 0.302941i
L(12)L(\frac12) ≈\approx 0.548917+0.302941i0.548917 + 0.302941i
L(32)L(\frac{3}{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)
bad2 1 1
13 1+(−3.23+1.59i)T 1 + (-3.23 + 1.59i)T
good3 1+(2.36+1.36i)T+(1.5+2.59i)T2 1 + (2.36 + 1.36i)T + (1.5 + 2.59i)T^{2}
5 1+(−0.633−0.633i)T+5iT2 1 + (-0.633 - 0.633i)T + 5iT^{2}
7 1+(−0.267+i)T+(−6.06−3.5i)T2 1 + (-0.267 + i)T + (-6.06 - 3.5i)T^{2}
11 1+(4.73−1.26i)T+(9.52−5.5i)T2 1 + (4.73 - 1.26i)T + (9.52 - 5.5i)T^{2}
17 1+(5.59−3.23i)T+(8.5−14.7i)T2 1 + (5.59 - 3.23i)T + (8.5 - 14.7i)T^{2}
19 1+(−3.09−0.830i)T+(16.4+9.5i)T2 1 + (-3.09 - 0.830i)T + (16.4 + 9.5i)T^{2}
23 1+(2.36−4.09i)T+(−11.5−19.9i)T2 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.5−2.59i)T+(−14.5−25.1i)T2 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2}
31 1+(−3.73+3.73i)T−31iT2 1 + (-3.73 + 3.73i)T - 31iT^{2}
37 1+(−2.86−10.6i)T+(−32.0+18.5i)T2 1 + (-2.86 - 10.6i)T + (-32.0 + 18.5i)T^{2}
41 1+(3.86−1.03i)T+(35.5−20.5i)T2 1 + (3.86 - 1.03i)T + (35.5 - 20.5i)T^{2}
43 1+(−1−1.73i)T+(−21.5+37.2i)T2 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2}
47 1+(−7.73−7.73i)T+47iT2 1 + (-7.73 - 7.73i)T + 47iT^{2}
53 1−7.73T+53T2 1 - 7.73T + 53T^{2}
59 1+(−0.464+1.73i)T+(−51.0−29.5i)T2 1 + (-0.464 + 1.73i)T + (-51.0 - 29.5i)T^{2}
61 1+(2.59+4.5i)T+(−30.5+52.8i)T2 1 + (2.59 + 4.5i)T + (-30.5 + 52.8i)T^{2}
67 1+(−3.36−12.5i)T+(−58.0+33.5i)T2 1 + (-3.36 - 12.5i)T + (-58.0 + 33.5i)T^{2}
71 1+(−2.36−0.633i)T+(61.4+35.5i)T2 1 + (-2.36 - 0.633i)T + (61.4 + 35.5i)T^{2}
73 1+(6.09−6.09i)T−73iT2 1 + (6.09 - 6.09i)T - 73iT^{2}
79 1−12.3iT−79T2 1 - 12.3iT - 79T^{2}
83 1+(−2.19+2.19i)T−83iT2 1 + (-2.19 + 2.19i)T - 83iT^{2}
89 1+(−1.90−7.09i)T+(−77.0+44.5i)T2 1 + (-1.90 - 7.09i)T + (-77.0 + 44.5i)T^{2}
97 1+(−0.437+1.63i)T+(−84.0−48.5i)T2 1 + (-0.437 + 1.63i)T + (-84.0 - 48.5i)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

−10.56905776234469501754943470461, −9.878214772040457316947107874886, −8.370444995412130037594570757844, −7.64921383491555513033059213511, −6.74649012842567371443288662366, −6.00650846145920164572244645303, −5.32250500447841829932095173491, −4.21922874103801157219867621943, −2.56296925648340434243643081461, −1.20535291398892866975302546881, 0.40660087288661687601838351112, 2.39191378306723175061536677328, 3.95153074090327286059396727939, 4.96866723654543807378392220498, 5.50973482323966908056885013825, 6.28792554110948242848040496863, 7.36039100459451827030183113297, 8.667672070933985001159675581802, 9.274740112214517414234903492204, 10.40377864107047625840351616772

Graph of the ZZ-function along the critical line