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Properties

Label 2-880-11.3-c1-0-23
Degree 22
Conductor 880880
Sign −0.259−0.965i-0.259 - 0.965i
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  + (−0.952 − 2.93i)3-s + (−0.809 − 0.587i)5-s + (−0.0119 + 0.0366i)7-s + (−5.25 + 3.81i)9-s + (1.11 − 3.12i)11-s + (−2.79 + 2.02i)13-s + (−0.952 + 2.93i)15-s + (−0.726 − 0.527i)17-s + (−0.373 − 1.14i)19-s + 0.118·21-s − 7.60·23-s + (0.309 + 0.951i)25-s + (8.71 + 6.33i)27-s + (−2.33 + 7.19i)29-s + (−4.81 + 3.50i)31-s + ⋯
L(s)  = 1  + (−0.549 − 1.69i)3-s + (−0.361 − 0.262i)5-s + (−0.00450 + 0.0138i)7-s + (−1.75 + 1.27i)9-s + (0.335 − 0.942i)11-s + (−0.774 + 0.562i)13-s + (−0.245 + 0.756i)15-s + (−0.176 − 0.127i)17-s + (−0.0855 − 0.263i)19-s + 0.0259·21-s − 1.58·23-s + (0.0618 + 0.190i)25-s + (1.67 + 1.21i)27-s + (−0.433 + 1.33i)29-s + (−0.865 + 0.628i)31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((−0.259−0.965i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=((−0.259−0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: −0.259−0.965i-0.259 - 0.965i
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), −0.259−0.965i)(2,\ 880,\ (\ :1/2),\ -0.259 - 0.965i)

Particular Values

L(1)L(1) ≈\approx 0.125466+0.163649i0.125466 + 0.163649i
L(12)L(\frac12) ≈\approx 0.125466+0.163649i0.125466 + 0.163649i
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+(−1.11+3.12i)T 1 + (-1.11 + 3.12i)T
good3 1+(0.952+2.93i)T+(−2.42+1.76i)T2 1 + (0.952 + 2.93i)T + (-2.42 + 1.76i)T^{2}
7 1+(0.0119−0.0366i)T+(−5.66−4.11i)T2 1 + (0.0119 - 0.0366i)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+(0.726+0.527i)T+(5.25+16.1i)T2 1 + (0.726 + 0.527i)T + (5.25 + 16.1i)T^{2}
19 1+(0.373+1.14i)T+(−15.3+11.1i)T2 1 + (0.373 + 1.14i)T + (-15.3 + 11.1i)T^{2}
23 1+7.60T+23T2 1 + 7.60T + 23T^{2}
29 1+(2.33−7.19i)T+(−23.4−17.0i)T2 1 + (2.33 - 7.19i)T + (-23.4 - 17.0i)T^{2}
31 1+(4.81−3.50i)T+(9.57−29.4i)T2 1 + (4.81 - 3.50i)T + (9.57 - 29.4i)T^{2}
37 1+(−3.31+10.1i)T+(−29.9−21.7i)T2 1 + (-3.31 + 10.1i)T + (-29.9 - 21.7i)T^{2}
41 1+(−0.954−2.93i)T+(−33.1+24.0i)T2 1 + (-0.954 - 2.93i)T + (-33.1 + 24.0i)T^{2}
43 1−9.68T+43T2 1 - 9.68T + 43T^{2}
47 1+(−0.403−1.24i)T+(−38.0+27.6i)T2 1 + (-0.403 - 1.24i)T + (-38.0 + 27.6i)T^{2}
53 1+(1.64−1.19i)T+(16.3−50.4i)T2 1 + (1.64 - 1.19i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.0573+0.176i)T+(−47.7−34.6i)T2 1 + (-0.0573 + 0.176i)T + (-47.7 - 34.6i)T^{2}
61 1+(−7.91−5.74i)T+(18.8+58.0i)T2 1 + (-7.91 - 5.74i)T + (18.8 + 58.0i)T^{2}
67 1+8.10T+67T2 1 + 8.10T + 67T^{2}
71 1+(5.81+4.22i)T+(21.9+67.5i)T2 1 + (5.81 + 4.22i)T + (21.9 + 67.5i)T^{2}
73 1+(−3.92+12.0i)T+(−59.0−42.9i)T2 1 + (-3.92 + 12.0i)T + (-59.0 - 42.9i)T^{2}
79 1+(10.2−7.44i)T+(24.4−75.1i)T2 1 + (10.2 - 7.44i)T + (24.4 - 75.1i)T^{2}
83 1+(8.69+6.31i)T+(25.6+78.9i)T2 1 + (8.69 + 6.31i)T + (25.6 + 78.9i)T^{2}
89 1+2.78T+89T2 1 + 2.78T + 89T^{2}
97 1+(14.3−10.4i)T+(29.9−92.2i)T2 1 + (14.3 - 10.4i)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

−9.282689665860721005572851252878, −8.555458208621525372841061879170, −7.56541535212102738521594992256, −7.12888696588134924034890920407, −6.10084784718361630770407259626, −5.47996562997889440079636291163, −4.11391611650145238586592619405, −2.60982137500247868389276070025, −1.46246299516013174752177307504, −0.10555039324007767530541063918, 2.49887652043534406384532308254, 3.92529837754976890383084669212, 4.28698035753948619857420773281, 5.37997198961931566176179767156, 6.14692426768127628210243259682, 7.38297366461981143627644533520, 8.319377081338024692227544422758, 9.487362733273115243243028833405, 9.928566909329621418113133225065, 10.50420070982577201853027315001

Graph of the ZZ-function along the critical line