Properties

Label 2-891-99.58-c1-0-31
Degree 22
Conductor 891891
Sign 0.7370.675i0.737 - 0.675i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.09 + 0.934i)2-s + (2.19 + 2.43i)4-s + (1.55 − 0.690i)5-s + (−0.203 + 0.0431i)7-s + (0.910 + 2.80i)8-s + 3.90·10-s + (2.04 − 2.60i)11-s + (−0.144 − 1.37i)13-s + (−0.466 − 0.0992i)14-s + (−0.0217 + 0.206i)16-s + (2.97 + 2.15i)17-s + (2.50 + 7.70i)19-s + (5.08 + 2.26i)20-s + (6.73 − 3.56i)22-s + (−1.99 + 3.45i)23-s + ⋯
L(s)  = 1  + (1.48 + 0.660i)2-s + (1.09 + 1.21i)4-s + (0.693 − 0.308i)5-s + (−0.0767 + 0.0163i)7-s + (0.321 + 0.990i)8-s + 1.23·10-s + (0.617 − 0.786i)11-s + (−0.0400 − 0.381i)13-s + (−0.124 − 0.0265i)14-s + (−0.00542 + 0.0516i)16-s + (0.720 + 0.523i)17-s + (0.574 + 1.76i)19-s + (1.13 + 0.506i)20-s + (1.43 − 0.759i)22-s + (−0.416 + 0.721i)23-s + ⋯

Functional equation

Λ(s)=(891s/2ΓC(s)L(s)=((0.7370.675i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(891s/2ΓC(s+1/2)L(s)=((0.7370.675i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 891891    =    34113^{4} \cdot 11
Sign: 0.7370.675i0.737 - 0.675i
Analytic conductor: 7.114677.11467
Root analytic conductor: 2.667332.66733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ891(190,)\chi_{891} (190, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 891, ( :1/2), 0.7370.675i)(2,\ 891,\ (\ :1/2),\ 0.737 - 0.675i)

Particular Values

L(1)L(1) \approx 3.74667+1.45669i3.74667 + 1.45669i
L(12)L(\frac12) \approx 3.74667+1.45669i3.74667 + 1.45669i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
11 1+(2.04+2.60i)T 1 + (-2.04 + 2.60i)T
good2 1+(2.090.934i)T+(1.33+1.48i)T2 1 + (-2.09 - 0.934i)T + (1.33 + 1.48i)T^{2}
5 1+(1.55+0.690i)T+(3.343.71i)T2 1 + (-1.55 + 0.690i)T + (3.34 - 3.71i)T^{2}
7 1+(0.2030.0431i)T+(6.392.84i)T2 1 + (0.203 - 0.0431i)T + (6.39 - 2.84i)T^{2}
13 1+(0.144+1.37i)T+(12.7+2.70i)T2 1 + (0.144 + 1.37i)T + (-12.7 + 2.70i)T^{2}
17 1+(2.972.15i)T+(5.25+16.1i)T2 1 + (-2.97 - 2.15i)T + (5.25 + 16.1i)T^{2}
19 1+(2.507.70i)T+(15.3+11.1i)T2 1 + (-2.50 - 7.70i)T + (-15.3 + 11.1i)T^{2}
23 1+(1.993.45i)T+(11.519.9i)T2 1 + (1.99 - 3.45i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.5300.112i)T+(26.411.7i)T2 1 + (0.530 - 0.112i)T + (26.4 - 11.7i)T^{2}
31 1+(1.02+9.77i)T+(30.3+6.44i)T2 1 + (1.02 + 9.77i)T + (-30.3 + 6.44i)T^{2}
37 1+(0.3811.17i)T+(29.921.7i)T2 1 + (0.381 - 1.17i)T + (-29.9 - 21.7i)T^{2}
41 1+(9.13+1.94i)T+(37.4+16.6i)T2 1 + (9.13 + 1.94i)T + (37.4 + 16.6i)T^{2}
43 1+(3.96+6.86i)T+(21.5+37.2i)T2 1 + (3.96 + 6.86i)T + (-21.5 + 37.2i)T^{2}
47 1+(5.255.83i)T+(4.9146.7i)T2 1 + (5.25 - 5.83i)T + (-4.91 - 46.7i)T^{2}
53 1+(5.303.85i)T+(16.350.4i)T2 1 + (5.30 - 3.85i)T + (16.3 - 50.4i)T^{2}
59 1+(5.265.84i)T+(6.16+58.6i)T2 1 + (-5.26 - 5.84i)T + (-6.16 + 58.6i)T^{2}
61 1+(0.145+1.38i)T+(59.612.6i)T2 1 + (-0.145 + 1.38i)T + (-59.6 - 12.6i)T^{2}
67 1+(2.92+5.06i)T+(33.558.0i)T2 1 + (-2.92 + 5.06i)T + (-33.5 - 58.0i)T^{2}
71 1+(10.37.49i)T+(21.9+67.5i)T2 1 + (-10.3 - 7.49i)T + (21.9 + 67.5i)T^{2}
73 1+(2.36+7.27i)T+(59.042.9i)T2 1 + (-2.36 + 7.27i)T + (-59.0 - 42.9i)T^{2}
79 1+(5.30+2.36i)T+(52.8+58.7i)T2 1 + (5.30 + 2.36i)T + (52.8 + 58.7i)T^{2}
83 1+(1.45+13.7i)T+(81.117.2i)T2 1 + (-1.45 + 13.7i)T + (-81.1 - 17.2i)T^{2}
89 1+6.19T+89T2 1 + 6.19T + 89T^{2}
97 1+(6.41+2.85i)T+(64.9+72.0i)T2 1 + (6.41 + 2.85i)T + (64.9 + 72.0i)T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)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

−10.10967024684845905861342676053, −9.502238422272909581961902565532, −8.198303965349340324822779579843, −7.54599599710368279524250782657, −6.26598472196005653006616284984, −5.83472643061899524898232460934, −5.20398337179677778083050404602, −3.85670447607201129126650059095, −3.33634410245590151166350592195, −1.65555379681884070845652021226, 1.64873074716557662719826053874, 2.66765120688604185935686543274, 3.57950605457240475530221167268, 4.79036417587050639679405463928, 5.22464122100216280999376401174, 6.59822860697523277573296913829, 6.81927020877516601420296670426, 8.415699260225294192664390608482, 9.614124654046048214090131294665, 10.08894674811143812754101736841

Graph of the ZZ-function along the critical line