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Properties

Label 2-2646-63.41-c1-0-5
Degree 22
Conductor 26462646
Sign −0.147−0.988i-0.147 - 0.988i
Analytic cond. 21.128421.1284
Root an. cond. 4.596564.59656
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.220 + 0.381i)5-s + 0.999i·8-s − 0.440i·10-s + (0.450 − 0.260i)11-s + (−5.55 − 3.20i)13-s + (−0.5 − 0.866i)16-s − 0.327·17-s − 4.23i·19-s + (0.220 + 0.381i)20-s + (−0.260 + 0.450i)22-s + (−1.25 − 0.725i)23-s + (2.40 + 4.16i)25-s + 6.40·26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0984 + 0.170i)5-s + 0.353i·8-s − 0.139i·10-s + (0.135 − 0.0784i)11-s + (−1.53 − 0.888i)13-s + (−0.125 − 0.216i)16-s − 0.0793·17-s − 0.972i·19-s + (0.0492 + 0.0852i)20-s + (−0.0554 + 0.0960i)22-s + (−0.261 − 0.151i)23-s + (0.480 + 0.832i)25-s + 1.25·26-s + ⋯

Functional equation

Λ(s)=(2646s/2 ΓC(s) L(s)=((−0.147−0.988i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2646s/2 ΓC(s+1/2) L(s)=((−0.147−0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26462646    =    2⋅33⋅722 \cdot 3^{3} \cdot 7^{2}
Sign: −0.147−0.988i-0.147 - 0.988i
Analytic conductor: 21.128421.1284
Root analytic conductor: 4.596564.59656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2646(881,⋅)\chi_{2646} (881, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2646, ( :1/2), −0.147−0.988i)(2,\ 2646,\ (\ :1/2),\ -0.147 - 0.988i)

Particular Values

L(1)L(1) ≈\approx 0.79596292560.7959629256
L(12)L(\frac12) ≈\approx 0.79596292560.7959629256
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+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
3 1 1
7 1 1
good5 1+(0.220−0.381i)T+(−2.5−4.33i)T2 1 + (0.220 - 0.381i)T + (-2.5 - 4.33i)T^{2}
11 1+(−0.450+0.260i)T+(5.5−9.52i)T2 1 + (-0.450 + 0.260i)T + (5.5 - 9.52i)T^{2}
13 1+(5.55+3.20i)T+(6.5+11.2i)T2 1 + (5.55 + 3.20i)T + (6.5 + 11.2i)T^{2}
17 1+0.327T+17T2 1 + 0.327T + 17T^{2}
19 1+4.23iT−19T2 1 + 4.23iT - 19T^{2}
23 1+(1.25+0.725i)T+(11.5+19.9i)T2 1 + (1.25 + 0.725i)T + (11.5 + 19.9i)T^{2}
29 1+(5.74−3.31i)T+(14.5−25.1i)T2 1 + (5.74 - 3.31i)T + (14.5 - 25.1i)T^{2}
31 1+(−6.07−3.50i)T+(15.5+26.8i)T2 1 + (-6.07 - 3.50i)T + (15.5 + 26.8i)T^{2}
37 1−3.68T+37T2 1 - 3.68T + 37T^{2}
41 1+(2.96−5.13i)T+(−20.5−35.5i)T2 1 + (2.96 - 5.13i)T + (-20.5 - 35.5i)T^{2}
43 1+(−5.21−9.03i)T+(−21.5+37.2i)T2 1 + (-5.21 - 9.03i)T + (-21.5 + 37.2i)T^{2}
47 1+(−4.02−6.97i)T+(−23.5+40.7i)T2 1 + (-4.02 - 6.97i)T + (-23.5 + 40.7i)T^{2}
53 1+7.95iT−53T2 1 + 7.95iT - 53T^{2}
59 1+(−2.45+4.24i)T+(−29.5−51.0i)T2 1 + (-2.45 + 4.24i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.33+0.771i)T+(30.5−52.8i)T2 1 + (-1.33 + 0.771i)T + (30.5 - 52.8i)T^{2}
67 1+(3.26−5.65i)T+(−33.5−58.0i)T2 1 + (3.26 - 5.65i)T + (-33.5 - 58.0i)T^{2}
71 1−16.2iT−71T2 1 - 16.2iT - 71T^{2}
73 1−4.12iT−73T2 1 - 4.12iT - 73T^{2}
79 1+(−0.662−1.14i)T+(−39.5+68.4i)T2 1 + (-0.662 - 1.14i)T + (-39.5 + 68.4i)T^{2}
83 1+(8.55+14.8i)T+(−41.5+71.8i)T2 1 + (8.55 + 14.8i)T + (-41.5 + 71.8i)T^{2}
89 1−11.7T+89T2 1 - 11.7T + 89T^{2}
97 1+(10.6−6.12i)T+(48.5−84.0i)T2 1 + (10.6 - 6.12i)T + (48.5 - 84.0i)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

−9.073305455695238834155179522908, −8.263973166307524637795692330475, −7.48640157600280249229949539770, −7.02688377114342051659684316278, −6.11514914682360265389513712817, −5.19891211894828725450672317433, −4.56768924318596683687947461123, −3.15914550039202470078754924680, −2.42524624421761607873348039778, −0.991096002444467319554657937452, 0.37479309074308555550342390143, 1.88083493192270947513988543608, 2.56463661055296688960908660503, 3.87306120605792020552423948196, 4.50459170890432988840462080517, 5.58445916397406422508507613925, 6.50748934038056720275358056536, 7.36012405839420032043295835476, 7.86180818683014211798337300717, 8.800539966619166823178958706936

Graph of the ZZ-function along the critical line