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Properties

Label 2-880-11.3-c1-0-4
Degree 22
Conductor 880880
Sign −0.941−0.335i-0.941 - 0.335i
Analytic cond. 7.026837.02683
Root an. cond. 2.650812.65081
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  + (0.868 + 2.67i)3-s + (0.809 + 0.587i)5-s + (−0.318 + 0.980i)7-s + (−3.96 + 2.88i)9-s + (−1.93 − 2.69i)11-s + (−2.79 + 2.02i)13-s + (−0.868 + 2.67i)15-s + (−1.94 − 1.40i)17-s + (2.36 + 7.29i)19-s − 2.89·21-s − 2.45·23-s + (0.309 + 0.951i)25-s + (−4.33 − 3.14i)27-s + (−1.83 + 5.66i)29-s + (−2.98 + 2.16i)31-s + ⋯
L(s)  = 1  + (0.501 + 1.54i)3-s + (0.361 + 0.262i)5-s + (−0.120 + 0.370i)7-s + (−1.32 + 0.961i)9-s + (−0.583 − 0.811i)11-s + (−0.773 + 0.562i)13-s + (−0.224 + 0.690i)15-s + (−0.470 − 0.341i)17-s + (0.543 + 1.67i)19-s − 0.632·21-s − 0.512·23-s + (0.0618 + 0.190i)25-s + (−0.833 − 0.605i)27-s + (−0.341 + 1.05i)29-s + (−0.535 + 0.389i)31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((−0.941−0.335i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=((−0.941−0.335i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: −0.941−0.335i-0.941 - 0.335i
Analytic conductor: 7.026837.02683
Root analytic conductor: 2.650812.65081
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ880(641,⋅)\chi_{880} (641, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 880, ( :1/2), −0.941−0.335i)(2,\ 880,\ (\ :1/2),\ -0.941 - 0.335i)

Particular Values

L(1)L(1) ≈\approx 0.251064+1.45100i0.251064 + 1.45100i
L(12)L(\frac12) ≈\approx 0.251064+1.45100i0.251064 + 1.45100i
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 1
5 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
11 1+(1.93+2.69i)T 1 + (1.93 + 2.69i)T
good3 1+(−0.868−2.67i)T+(−2.42+1.76i)T2 1 + (-0.868 - 2.67i)T + (-2.42 + 1.76i)T^{2}
7 1+(0.318−0.980i)T+(−5.66−4.11i)T2 1 + (0.318 - 0.980i)T + (-5.66 - 4.11i)T^{2}
13 1+(2.79−2.02i)T+(4.01−12.3i)T2 1 + (2.79 - 2.02i)T + (4.01 - 12.3i)T^{2}
17 1+(1.94+1.40i)T+(5.25+16.1i)T2 1 + (1.94 + 1.40i)T + (5.25 + 16.1i)T^{2}
19 1+(−2.36−7.29i)T+(−15.3+11.1i)T2 1 + (-2.36 - 7.29i)T + (-15.3 + 11.1i)T^{2}
23 1+2.45T+23T2 1 + 2.45T + 23T^{2}
29 1+(1.83−5.66i)T+(−23.4−17.0i)T2 1 + (1.83 - 5.66i)T + (-23.4 - 17.0i)T^{2}
31 1+(2.98−2.16i)T+(9.57−29.4i)T2 1 + (2.98 - 2.16i)T + (9.57 - 29.4i)T^{2}
37 1+(−1.84+5.66i)T+(−29.9−21.7i)T2 1 + (-1.84 + 5.66i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.21−3.74i)T+(−33.1+24.0i)T2 1 + (-1.21 - 3.74i)T + (-33.1 + 24.0i)T^{2}
43 1−7.64T+43T2 1 - 7.64T + 43T^{2}
47 1+(1.80+5.55i)T+(−38.0+27.6i)T2 1 + (1.80 + 5.55i)T + (-38.0 + 27.6i)T^{2}
53 1+(−9.58+6.96i)T+(16.3−50.4i)T2 1 + (-9.58 + 6.96i)T + (16.3 - 50.4i)T^{2}
59 1+(0.910−2.80i)T+(−47.7−34.6i)T2 1 + (0.910 - 2.80i)T + (-47.7 - 34.6i)T^{2}
61 1+(2.00+1.45i)T+(18.8+58.0i)T2 1 + (2.00 + 1.45i)T + (18.8 + 58.0i)T^{2}
67 1−6.14T+67T2 1 - 6.14T + 67T^{2}
71 1+(−1.63−1.18i)T+(21.9+67.5i)T2 1 + (-1.63 - 1.18i)T + (21.9 + 67.5i)T^{2}
73 1+(0.255−0.785i)T+(−59.0−42.9i)T2 1 + (0.255 - 0.785i)T + (-59.0 - 42.9i)T^{2}
79 1+(9.77−7.09i)T+(24.4−75.1i)T2 1 + (9.77 - 7.09i)T + (24.4 - 75.1i)T^{2}
83 1+(−1.30−0.946i)T+(25.6+78.9i)T2 1 + (-1.30 - 0.946i)T + (25.6 + 78.9i)T^{2}
89 1−8.16T+89T2 1 - 8.16T + 89T^{2}
97 1+(1.97−1.43i)T+(29.9−92.2i)T2 1 + (1.97 - 1.43i)T + (29.9 - 92.2i)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

−10.37011072474786469907015110235, −9.670727629197267511422265890478, −9.062771064219477536108743239902, −8.267113648740554380299992445268, −7.20772259636429414698646626657, −5.81374940264983587276769085223, −5.25777701286262496021250972736, −4.11172224995221627346444682822, −3.27869797593987076232954429755, −2.26810184678963567269061836016, 0.63046740337158463986626242252, 2.10379146800534435647610295489, 2.74678421671360840190480098910, 4.39989256130125828170721038239, 5.52868370431480361192987219402, 6.54458704235128869802665333661, 7.40785715075132343221326848641, 7.73810518220014245695593135342, 8.837954326806085534444966365795, 9.618804272841441442680022123739

Graph of the ZZ-function along the critical line