Properties

Label 2-891-99.25-c1-0-17
Degree 22
Conductor 891891
Sign 0.1770.984i-0.177 - 0.984i
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  + (0.0330 + 0.314i)2-s + (1.85 − 0.395i)4-s + (−0.423 + 4.03i)5-s + (2.28 + 2.53i)7-s + (0.380 + 1.17i)8-s − 1.28·10-s + (2.11 + 2.55i)11-s + (1.26 + 0.562i)13-s + (−0.721 + 0.801i)14-s + (3.11 − 1.38i)16-s + (−5.20 − 3.78i)17-s + (−1.07 − 3.31i)19-s + (0.805 + 7.66i)20-s + (−0.733 + 0.749i)22-s + (−1.86 − 3.23i)23-s + ⋯
L(s)  = 1  + (0.0233 + 0.222i)2-s + (0.929 − 0.197i)4-s + (−0.189 + 1.80i)5-s + (0.862 + 0.957i)7-s + (0.134 + 0.414i)8-s − 0.405·10-s + (0.637 + 0.770i)11-s + (0.350 + 0.155i)13-s + (−0.192 + 0.214i)14-s + (0.778 − 0.346i)16-s + (−1.26 − 0.916i)17-s + (−0.247 − 0.760i)19-s + (0.180 + 1.71i)20-s + (−0.156 + 0.159i)22-s + (−0.389 − 0.675i)23-s + ⋯

Functional equation

Λ(s)=(891s/2ΓC(s)L(s)=((0.1770.984i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(891s/2ΓC(s+1/2)L(s)=((0.1770.984i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 891891    =    34113^{4} \cdot 11
Sign: 0.1770.984i-0.177 - 0.984i
Analytic conductor: 7.114677.11467
Root analytic conductor: 2.667332.66733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ891(784,)\chi_{891} (784, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 891, ( :1/2), 0.1770.984i)(2,\ 891,\ (\ :1/2),\ -0.177 - 0.984i)

Particular Values

L(1)L(1) \approx 1.35136+1.61759i1.35136 + 1.61759i
L(12)L(\frac12) \approx 1.35136+1.61759i1.35136 + 1.61759i
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+(2.112.55i)T 1 + (-2.11 - 2.55i)T
good2 1+(0.03300.314i)T+(1.95+0.415i)T2 1 + (-0.0330 - 0.314i)T + (-1.95 + 0.415i)T^{2}
5 1+(0.4234.03i)T+(4.891.03i)T2 1 + (0.423 - 4.03i)T + (-4.89 - 1.03i)T^{2}
7 1+(2.282.53i)T+(0.731+6.96i)T2 1 + (-2.28 - 2.53i)T + (-0.731 + 6.96i)T^{2}
13 1+(1.260.562i)T+(8.69+9.66i)T2 1 + (-1.26 - 0.562i)T + (8.69 + 9.66i)T^{2}
17 1+(5.20+3.78i)T+(5.25+16.1i)T2 1 + (5.20 + 3.78i)T + (5.25 + 16.1i)T^{2}
19 1+(1.07+3.31i)T+(15.3+11.1i)T2 1 + (1.07 + 3.31i)T + (-15.3 + 11.1i)T^{2}
23 1+(1.86+3.23i)T+(11.5+19.9i)T2 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.0499+0.0554i)T+(3.03+28.8i)T2 1 + (0.0499 + 0.0554i)T + (-3.03 + 28.8i)T^{2}
31 1+(3.41+1.52i)T+(20.7+23.0i)T2 1 + (3.41 + 1.52i)T + (20.7 + 23.0i)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+(6.23+6.92i)T+(4.2840.7i)T2 1 + (-6.23 + 6.92i)T + (-4.28 - 40.7i)T^{2}
43 1+(0.228+0.395i)T+(21.537.2i)T2 1 + (-0.228 + 0.395i)T + (-21.5 - 37.2i)T^{2}
47 1+(2.140.456i)T+(42.9+19.1i)T2 1 + (-2.14 - 0.456i)T + (42.9 + 19.1i)T^{2}
53 1+(0.729+0.530i)T+(16.350.4i)T2 1 + (-0.729 + 0.530i)T + (16.3 - 50.4i)T^{2}
59 1+(7.84+1.66i)T+(53.823.9i)T2 1 + (-7.84 + 1.66i)T + (53.8 - 23.9i)T^{2}
61 1+(5.27+2.34i)T+(40.845.3i)T2 1 + (-5.27 + 2.34i)T + (40.8 - 45.3i)T^{2}
67 1+(2.925.06i)T+(33.5+58.0i)T2 1 + (-2.92 - 5.06i)T + (-33.5 + 58.0i)T^{2}
71 1+(7.665.56i)T+(21.9+67.5i)T2 1 + (-7.66 - 5.56i)T + (21.9 + 67.5i)T^{2}
73 1+(4.34+13.3i)T+(59.042.9i)T2 1 + (-4.34 + 13.3i)T + (-59.0 - 42.9i)T^{2}
79 1+(1.1410.9i)T+(77.2+16.4i)T2 1 + (-1.14 - 10.9i)T + (-77.2 + 16.4i)T^{2}
83 1+(1.240.556i)T+(55.561.6i)T2 1 + (1.24 - 0.556i)T + (55.5 - 61.6i)T^{2}
89 1+10.3T+89T2 1 + 10.3T + 89T^{2}
97 1+(0.8628.20i)T+(94.8+20.1i)T2 1 + (-0.862 - 8.20i)T + (-94.8 + 20.1i)T^{2}
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   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

−10.63918531720483636255908670847, −9.600737831240384404485081963836, −8.590286442268777293937036322865, −7.50974469257793913183993660902, −6.85334176598138516949563543144, −6.37460816742549666309175196742, −5.26620668212842863655888525404, −3.97709971939137004679562882907, −2.47885661618552087298043627925, −2.21058739083045129476485213465, 1.05973239339167738258236659489, 1.81713596381233582483468384259, 3.80048538623248192805025457151, 4.24416071049026031512706242629, 5.49225556717972759420145268425, 6.37213039182549431993631535542, 7.57327473417837125750663133940, 8.253031642245758603927057083607, 8.823649820796953535378813767867, 9.956578039245776300568890379081

Graph of the ZZ-function along the critical line