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Properties

Label 2-1254-209.65-c1-0-12
Degree 22
Conductor 12541254
Sign 0.841−0.540i0.841 - 0.540i
Analytic cond. 10.013210.0132
Root an. cond. 3.164373.16437
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  + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.704 + 1.22i)5-s + (0.866 − 0.499i)6-s − 1.83i·7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.704 + 1.22i)10-s + (−2.01 + 2.63i)11-s − 0.999i·12-s + (2.83 + 4.91i)13-s + (−1.58 − 0.915i)14-s + (−1.22 + 0.704i)15-s + (−0.5 + 0.866i)16-s + (2.61 + 1.51i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.315 + 0.545i)5-s + (0.353 − 0.204i)6-s − 0.692i·7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.222 + 0.385i)10-s + (−0.606 + 0.794i)11-s − 0.288i·12-s + (0.786 + 1.36i)13-s + (−0.423 − 0.244i)14-s + (−0.315 + 0.181i)15-s + (−0.125 + 0.216i)16-s + (0.634 + 0.366i)17-s + ⋯

Functional equation

Λ(s)=(1254s/2 ΓC(s) L(s)=((0.841−0.540i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1254s/2 ΓC(s+1/2) L(s)=((0.841−0.540i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12541254    =    2⋅3⋅11⋅192 \cdot 3 \cdot 11 \cdot 19
Sign: 0.841−0.540i0.841 - 0.540i
Analytic conductor: 10.013210.0132
Root analytic conductor: 3.164373.16437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1254(901,⋅)\chi_{1254} (901, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1254, ( :1/2), 0.841−0.540i)(2,\ 1254,\ (\ :1/2),\ 0.841 - 0.540i)

Particular Values

L(1)L(1) ≈\approx 1.9940437931.994043793
L(12)L(\frac12) ≈\approx 1.9940437931.994043793
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
11 1+(2.01−2.63i)T 1 + (2.01 - 2.63i)T
19 1+(−0.339−4.34i)T 1 + (-0.339 - 4.34i)T
good5 1+(0.704−1.22i)T+(−2.5−4.33i)T2 1 + (0.704 - 1.22i)T + (-2.5 - 4.33i)T^{2}
7 1+1.83iT−7T2 1 + 1.83iT - 7T^{2}
13 1+(−2.83−4.91i)T+(−6.5+11.2i)T2 1 + (-2.83 - 4.91i)T + (-6.5 + 11.2i)T^{2}
17 1+(−2.61−1.51i)T+(8.5+14.7i)T2 1 + (-2.61 - 1.51i)T + (8.5 + 14.7i)T^{2}
23 1+(4.52+7.84i)T+(−11.5+19.9i)T2 1 + (4.52 + 7.84i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.59−4.48i)T+(−14.5+25.1i)T2 1 + (-2.59 - 4.48i)T + (-14.5 + 25.1i)T^{2}
31 1−7.54iT−31T2 1 - 7.54iT - 31T^{2}
37 1+3.85iT−37T2 1 + 3.85iT - 37T^{2}
41 1+(−4.68+8.11i)T+(−20.5−35.5i)T2 1 + (-4.68 + 8.11i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.31+0.758i)T+(21.5+37.2i)T2 1 + (1.31 + 0.758i)T + (21.5 + 37.2i)T^{2}
47 1+(−3.82−6.62i)T+(−23.5+40.7i)T2 1 + (-3.82 - 6.62i)T + (-23.5 + 40.7i)T^{2}
53 1+(−8.05+4.64i)T+(26.5−45.8i)T2 1 + (-8.05 + 4.64i)T + (26.5 - 45.8i)T^{2}
59 1+(5.58+3.22i)T+(29.5+51.0i)T2 1 + (5.58 + 3.22i)T + (29.5 + 51.0i)T^{2}
61 1+(1.55−0.896i)T+(30.5−52.8i)T2 1 + (1.55 - 0.896i)T + (30.5 - 52.8i)T^{2}
67 1+(−1.64+0.949i)T+(33.5−58.0i)T2 1 + (-1.64 + 0.949i)T + (33.5 - 58.0i)T^{2}
71 1+(7.63+4.40i)T+(35.5+61.4i)T2 1 + (7.63 + 4.40i)T + (35.5 + 61.4i)T^{2}
73 1+(−12.3−7.10i)T+(36.5+63.2i)T2 1 + (-12.3 - 7.10i)T + (36.5 + 63.2i)T^{2}
79 1+(6.73−11.6i)T+(−39.5−68.4i)T2 1 + (6.73 - 11.6i)T + (-39.5 - 68.4i)T^{2}
83 1−7.43iT−83T2 1 - 7.43iT - 83T^{2}
89 1+(14.6−8.48i)T+(44.5−77.0i)T2 1 + (14.6 - 8.48i)T + (44.5 - 77.0i)T^{2}
97 1+(1.45+0.840i)T+(48.5+84.0i)T2 1 + (1.45 + 0.840i)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

−10.05269553546613801552267121495, −9.039446105839490070473441930302, −8.250080726141682270201511911689, −7.26792936330891308720577051122, −6.56406843684995831775325230670, −5.34229122160726388069668504163, −4.17427275256211763743618503404, −3.81762661448249570383311795630, −2.63442305991333540898123380211, −1.51849072068360904255490633974, 0.74728100737046351657581015340, 2.63179031269754108773338433062, 3.40381814772007513259447951036, 4.55282024211947431600824140395, 5.70506167281648831275020611649, 5.94641795377485598276108776111, 7.41141771948535847522641614310, 8.034995206096461889093340628951, 8.497632892922709367942619702861, 9.354030871006197746466188006846

Graph of the ZZ-function along the critical line