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Properties

Label 2-2646-63.20-c1-0-35
Degree 22
Conductor 26462646
Sign −0.999+0.00551i-0.999 + 0.00551i
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 + (−1.77 − 3.07i)5-s − 0.999i·8-s + 3.55i·10-s + (2.61 + 1.51i)11-s + (−0.888 + 0.513i)13-s + (−0.5 + 0.866i)16-s − 1.61·17-s − 8.22i·19-s + (1.77 − 3.07i)20-s + (−1.51 − 2.61i)22-s + (2.90 − 1.67i)23-s + (−3.80 + 6.59i)25-s + 1.02·26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.794 − 1.37i)5-s − 0.353i·8-s + 1.12i·10-s + (0.789 + 0.455i)11-s + (−0.246 + 0.142i)13-s + (−0.125 + 0.216i)16-s − 0.392·17-s − 1.88i·19-s + (0.397 − 0.687i)20-s + (−0.322 − 0.558i)22-s + (0.606 − 0.350i)23-s + (−0.761 + 1.31i)25-s + 0.201·26-s + ⋯

Functional equation

Λ(s)=(2646s/2 ΓC(s) L(s)=((−0.999+0.00551i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00551i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2646s/2 ΓC(s+1/2) L(s)=((−0.999+0.00551i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00551i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26462646    =    2⋅33⋅722 \cdot 3^{3} \cdot 7^{2}
Sign: −0.999+0.00551i-0.999 + 0.00551i
Analytic conductor: 21.128421.1284
Root analytic conductor: 4.596564.59656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2646(1763,⋅)\chi_{2646} (1763, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2646, ( :1/2), −0.999+0.00551i)(2,\ 2646,\ (\ :1/2),\ -0.999 + 0.00551i)

Particular Values

L(1)L(1) ≈\approx 0.68726759260.6872675926
L(12)L(\frac12) ≈\approx 0.68726759260.6872675926
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+(1.77+3.07i)T+(−2.5+4.33i)T2 1 + (1.77 + 3.07i)T + (-2.5 + 4.33i)T^{2}
11 1+(−2.61−1.51i)T+(5.5+9.52i)T2 1 + (-2.61 - 1.51i)T + (5.5 + 9.52i)T^{2}
13 1+(0.888−0.513i)T+(6.5−11.2i)T2 1 + (0.888 - 0.513i)T + (6.5 - 11.2i)T^{2}
17 1+1.61T+17T2 1 + 1.61T + 17T^{2}
19 1+8.22iT−19T2 1 + 8.22iT - 19T^{2}
23 1+(−2.90+1.67i)T+(11.5−19.9i)T2 1 + (-2.90 + 1.67i)T + (11.5 - 19.9i)T^{2}
29 1+(−3.70−2.13i)T+(14.5+25.1i)T2 1 + (-3.70 - 2.13i)T + (14.5 + 25.1i)T^{2}
31 1+(−5.18+2.99i)T+(15.5−26.8i)T2 1 + (-5.18 + 2.99i)T + (15.5 - 26.8i)T^{2}
37 1+5.84T+37T2 1 + 5.84T + 37T^{2}
41 1+(0.0472+0.0817i)T+(−20.5+35.5i)T2 1 + (0.0472 + 0.0817i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3.05+5.29i)T+(−21.5−37.2i)T2 1 + (-3.05 + 5.29i)T + (-21.5 - 37.2i)T^{2}
47 1+(2.57−4.45i)T+(−23.5−40.7i)T2 1 + (2.57 - 4.45i)T + (-23.5 - 40.7i)T^{2}
53 1+3.18iT−53T2 1 + 3.18iT - 53T^{2}
59 1+(4.42+7.65i)T+(−29.5+51.0i)T2 1 + (4.42 + 7.65i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.06+2.34i)T+(30.5+52.8i)T2 1 + (4.06 + 2.34i)T + (30.5 + 52.8i)T^{2}
67 1+(−0.187−0.325i)T+(−33.5+58.0i)T2 1 + (-0.187 - 0.325i)T + (-33.5 + 58.0i)T^{2}
71 1+13.9iT−71T2 1 + 13.9iT - 71T^{2}
73 1+1.31iT−73T2 1 + 1.31iT - 73T^{2}
79 1+(0.462−0.800i)T+(−39.5−68.4i)T2 1 + (0.462 - 0.800i)T + (-39.5 - 68.4i)T^{2}
83 1+(5.43−9.40i)T+(−41.5−71.8i)T2 1 + (5.43 - 9.40i)T + (-41.5 - 71.8i)T^{2}
89 1+4.70T+89T2 1 + 4.70T + 89T^{2}
97 1+(−13.3−7.69i)T+(48.5+84.0i)T2 1 + (-13.3 - 7.69i)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

−8.738832457967267998931709563985, −7.911660081542749768162392990247, −7.09133408721228240691676251498, −6.43997329341671585454187722564, −4.89453322300004731183117159726, −4.66811029195706192022489137416, −3.65633408770352291387157815747, −2.48640013672105739723492275378, −1.26116240922118748072791123922, −0.31590253289337403104240652563, 1.35790983184296411112198020882, 2.73488289944109711356701629690, 3.50692652192014154239736502235, 4.37542831837878270118289853150, 5.69072033252672201666627758647, 6.41941446036274708077623799022, 6.98732246177419610402113678100, 7.72949529324397990396931106037, 8.331809494285850705906967445482, 9.123667651715400272902786576829

Graph of the ZZ-function along the critical line