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Properties

Label 2-1254-19.11-c1-0-10
Degree 22
Conductor 12541254
Sign 0.813−0.582i0.813 - 0.582i
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 + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 7-s − 0.999·8-s + (−0.499 − 0.866i)9-s − 11-s + 0.999·12-s + (2 + 3.46i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (1.5 − 2.59i)17-s − 0.999·18-s + (−0.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.204 + 0.353i)6-s − 0.377·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s − 0.301·11-s + 0.288·12-s + (0.554 + 0.960i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.363 − 0.630i)17-s − 0.235·18-s + (−0.114 + 0.993i)19-s + ⋯

Functional equation

Λ(s)=(1254s/2 ΓC(s) L(s)=((0.813−0.582i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1254s/2 ΓC(s+1/2) L(s)=((0.813−0.582i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12541254    =    2⋅3⋅11⋅192 \cdot 3 \cdot 11 \cdot 19
Sign: 0.813−0.582i0.813 - 0.582i
Analytic conductor: 10.013210.0132
Root analytic conductor: 3.164373.16437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1254(1189,⋅)\chi_{1254} (1189, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1254, ( :1/2), 0.813−0.582i)(2,\ 1254,\ (\ :1/2),\ 0.813 - 0.582i)

Particular Values

L(1)L(1) ≈\approx 1.4393069461.439306946
L(12)L(\frac12) ≈\approx 1.4393069461.439306946
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
11 1+T 1 + T
19 1+(0.5−4.33i)T 1 + (0.5 - 4.33i)T
good5 1+(−2.5−4.33i)T2 1 + (-2.5 - 4.33i)T^{2}
7 1+T+7T2 1 + T + 7T^{2}
13 1+(−2−3.46i)T+(−6.5+11.2i)T2 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2}
17 1+(−1.5+2.59i)T+(−8.5−14.7i)T2 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2}
23 1+(1.5+2.59i)T+(−11.5+19.9i)T2 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2}
29 1+(−4.5−7.79i)T+(−14.5+25.1i)T2 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2}
31 1−2T+31T2 1 - 2T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+(3−5.19i)T+(−20.5−35.5i)T2 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.5+6.06i)T+(−21.5−37.2i)T2 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2}
47 1+(−1.5−2.59i)T+(−23.5+40.7i)T2 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2}
53 1+(−6−10.3i)T+(−26.5+45.8i)T2 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2}
59 1+(−29.5−51.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(−5−8.66i)T+(−30.5+52.8i)T2 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2−3.46i)T+(−33.5+58.0i)T2 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2}
71 1+(−4.5+7.79i)T+(−35.5−61.4i)T2 1 + (-4.5 + 7.79i)T + (-35.5 - 61.4i)T^{2}
73 1+(1−1.73i)T+(−36.5−63.2i)T2 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2}
79 1+(4−6.92i)T+(−39.5−68.4i)T2 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1+(3+5.19i)T+(−44.5+77.0i)T2 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2}
97 1+(5.5−9.52i)T+(−48.5−84.0i)T2 1 + (5.5 - 9.52i)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

−9.937615253785856090071833122571, −9.149266971101737756660440138579, −8.425887385706156858017927079795, −7.14704104904295897991657053997, −6.28269199112625360851071597890, −5.41863377542927728609171288729, −4.53736144114000981193898448338, −3.66740614331736106506027672789, −2.74253233876165412333322618281, −1.27075126801111658495873671764, 0.62644032408619753879932427441, 2.44551273958970450907355079014, 3.50799744926839791682923743595, 4.64360159764388942719442269385, 5.59900790549217667417471231787, 6.28025574424133776786204982113, 6.97951230279840894964243277712, 8.077084450076726766986258696292, 8.349909152167721560154844690951, 9.620001934421596755830855360690

Graph of the ZZ-function along the critical line