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Properties

Label 2-891-11.3-c1-0-13
Degree 22
Conductor 891891
Sign 0.698−0.715i0.698 - 0.715i
Analytic cond. 7.114677.11467
Root an. cond. 2.667332.66733
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  + (1.48 − 1.07i)2-s + (0.421 − 1.29i)4-s + (0.0729 + 0.0530i)5-s + (−1.45 + 4.48i)7-s + (0.360 + 1.11i)8-s + 0.165·10-s + (−3.00 − 1.39i)11-s + (−2.65 + 1.92i)13-s + (2.67 + 8.21i)14-s + (3.93 + 2.86i)16-s + (3.80 + 2.76i)17-s + (1.79 + 5.53i)19-s + (0.0995 − 0.0723i)20-s + (−5.96 + 1.16i)22-s − 2.20·23-s + ⋯
L(s)  = 1  + (1.04 − 0.762i)2-s + (0.210 − 0.648i)4-s + (0.0326 + 0.0237i)5-s + (−0.550 + 1.69i)7-s + (0.127 + 0.392i)8-s + 0.0523·10-s + (−0.906 − 0.421i)11-s + (−0.735 + 0.534i)13-s + (0.713 + 2.19i)14-s + (0.984 + 0.715i)16-s + (0.924 + 0.671i)17-s + (0.412 + 1.27i)19-s + (0.0222 − 0.0161i)20-s + (−1.27 + 0.248i)22-s − 0.458·23-s + ⋯

Functional equation

Λ(s)=(891s/2 ΓC(s) L(s)=((0.698−0.715i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(891s/2 ΓC(s+1/2) L(s)=((0.698−0.715i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 891891    =    34⋅113^{4} \cdot 11
Sign: 0.698−0.715i0.698 - 0.715i
Analytic conductor: 7.114677.11467
Root analytic conductor: 2.667332.66733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ891(487,⋅)\chi_{891} (487, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 891, ( :1/2), 0.698−0.715i)(2,\ 891,\ (\ :1/2),\ 0.698 - 0.715i)

Particular Values

L(1)L(1) ≈\approx 2.00257+0.843580i2.00257 + 0.843580i
L(12)L(\frac12) ≈\approx 2.00257+0.843580i2.00257 + 0.843580i
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)
bad3 1 1
11 1+(3.00+1.39i)T 1 + (3.00 + 1.39i)T
good2 1+(−1.48+1.07i)T+(0.618−1.90i)T2 1 + (-1.48 + 1.07i)T + (0.618 - 1.90i)T^{2}
5 1+(−0.0729−0.0530i)T+(1.54+4.75i)T2 1 + (-0.0729 - 0.0530i)T + (1.54 + 4.75i)T^{2}
7 1+(1.45−4.48i)T+(−5.66−4.11i)T2 1 + (1.45 - 4.48i)T + (-5.66 - 4.11i)T^{2}
13 1+(2.65−1.92i)T+(4.01−12.3i)T2 1 + (2.65 - 1.92i)T + (4.01 - 12.3i)T^{2}
17 1+(−3.80−2.76i)T+(5.25+16.1i)T2 1 + (-3.80 - 2.76i)T + (5.25 + 16.1i)T^{2}
19 1+(−1.79−5.53i)T+(−15.3+11.1i)T2 1 + (-1.79 - 5.53i)T + (-15.3 + 11.1i)T^{2}
23 1+2.20T+23T2 1 + 2.20T + 23T^{2}
29 1+(−1.57+4.85i)T+(−23.4−17.0i)T2 1 + (-1.57 + 4.85i)T + (-23.4 - 17.0i)T^{2}
31 1+(−3.95+2.87i)T+(9.57−29.4i)T2 1 + (-3.95 + 2.87i)T + (9.57 - 29.4i)T^{2}
37 1+(1.41−4.34i)T+(−29.9−21.7i)T2 1 + (1.41 - 4.34i)T + (-29.9 - 21.7i)T^{2}
41 1+(−2.32−7.14i)T+(−33.1+24.0i)T2 1 + (-2.32 - 7.14i)T + (-33.1 + 24.0i)T^{2}
43 1+5.79T+43T2 1 + 5.79T + 43T^{2}
47 1+(0.290+0.893i)T+(−38.0+27.6i)T2 1 + (0.290 + 0.893i)T + (-38.0 + 27.6i)T^{2}
53 1+(−9.93+7.21i)T+(16.3−50.4i)T2 1 + (-9.93 + 7.21i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.195+0.602i)T+(−47.7−34.6i)T2 1 + (-0.195 + 0.602i)T + (-47.7 - 34.6i)T^{2}
61 1+(−0.944−0.686i)T+(18.8+58.0i)T2 1 + (-0.944 - 0.686i)T + (18.8 + 58.0i)T^{2}
67 1−4.12T+67T2 1 - 4.12T + 67T^{2}
71 1+(−6.15−4.47i)T+(21.9+67.5i)T2 1 + (-6.15 - 4.47i)T + (21.9 + 67.5i)T^{2}
73 1+(−3.24+9.98i)T+(−59.0−42.9i)T2 1 + (-3.24 + 9.98i)T + (-59.0 - 42.9i)T^{2}
79 1+(7.27−5.28i)T+(24.4−75.1i)T2 1 + (7.27 - 5.28i)T + (24.4 - 75.1i)T^{2}
83 1+(−11.4−8.33i)T+(25.6+78.9i)T2 1 + (-11.4 - 8.33i)T + (25.6 + 78.9i)T^{2}
89 1−2.58T+89T2 1 - 2.58T + 89T^{2}
97 1+(−4.00+2.91i)T+(29.9−92.2i)T2 1 + (-4.00 + 2.91i)T + (29.9 - 92.2i)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

−10.12102116439718185163999036927, −9.820072462878761085733681494700, −8.337360247025792520191843321808, −8.039062794233897692036551266131, −6.26846737539008145249290779449, −5.70705637130474923475037826068, −4.93209784397747121603002043669, −3.71875131675853201981242488111, −2.77248711273341353606849046854, −2.06624439696551477434023977932, 0.72886308652330044803633918612, 2.94921119851478350248338868027, 3.83204835558636786204875861402, 4.90245408072371147017065330791, 5.38001688536976330923667487950, 6.70457356501171026564431877291, 7.33376017061547672281689872214, 7.68686340239109680975688756988, 9.338454352006755947392253180220, 10.19637202007502149669527355479

Graph of the ZZ-function along the critical line