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Properties

Label 2-42e2-9.4-c1-0-8
Degree 22
Conductor 17641764
Sign 0.449−0.893i0.449 - 0.893i
Analytic cond. 14.085614.0856
Root an. cond. 3.753083.75308
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  + (−1.64 − 0.545i)3-s + (−0.849 + 1.47i)5-s + (2.40 + 1.79i)9-s + (−1.23 − 2.14i)11-s + (0.388 − 0.673i)13-s + (2.19 − 1.95i)15-s − 2.81·17-s + 4.98·19-s + (−0.356 + 0.616i)23-s + (1.05 + 1.82i)25-s + (−2.97 − 4.25i)27-s + (−2.25 − 3.90i)29-s + (2.54 − 4.41i)31-s + (0.866 + 4.20i)33-s − 6.87·37-s + ⋯
L(s)  = 1  + (−0.949 − 0.314i)3-s + (−0.380 + 0.658i)5-s + (0.801 + 0.597i)9-s + (−0.373 − 0.646i)11-s + (0.107 − 0.186i)13-s + (0.567 − 0.505i)15-s − 0.681·17-s + 1.14·19-s + (−0.0742 + 0.128i)23-s + (0.211 + 0.365i)25-s + (−0.572 − 0.819i)27-s + (−0.418 − 0.725i)29-s + (0.457 − 0.793i)31-s + (0.150 + 0.731i)33-s − 1.13·37-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((0.449−0.893i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s+1/2) L(s)=((0.449−0.893i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.449−0.893i0.449 - 0.893i
Analytic conductor: 14.085614.0856
Root analytic conductor: 3.753083.75308
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1764(589,⋅)\chi_{1764} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :1/2), 0.449−0.893i)(2,\ 1764,\ (\ :1/2),\ 0.449 - 0.893i)

Particular Values

L(1)L(1) ≈\approx 0.86395344330.8639534433
L(12)L(\frac12) ≈\approx 0.86395344330.8639534433
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
3 1+(1.64+0.545i)T 1 + (1.64 + 0.545i)T
7 1 1
good5 1+(0.849−1.47i)T+(−2.5−4.33i)T2 1 + (0.849 - 1.47i)T + (-2.5 - 4.33i)T^{2}
11 1+(1.23+2.14i)T+(−5.5+9.52i)T2 1 + (1.23 + 2.14i)T + (-5.5 + 9.52i)T^{2}
13 1+(−0.388+0.673i)T+(−6.5−11.2i)T2 1 + (-0.388 + 0.673i)T + (-6.5 - 11.2i)T^{2}
17 1+2.81T+17T2 1 + 2.81T + 17T^{2}
19 1−4.98T+19T2 1 - 4.98T + 19T^{2}
23 1+(0.356−0.616i)T+(−11.5−19.9i)T2 1 + (0.356 - 0.616i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.25+3.90i)T+(−14.5+25.1i)T2 1 + (2.25 + 3.90i)T + (-14.5 + 25.1i)T^{2}
31 1+(−2.54+4.41i)T+(−15.5−26.8i)T2 1 + (-2.54 + 4.41i)T + (-15.5 - 26.8i)T^{2}
37 1+6.87T+37T2 1 + 6.87T + 37T^{2}
41 1+(2.93−5.08i)T+(−20.5−35.5i)T2 1 + (2.93 - 5.08i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.32−4.03i)T+(−21.5+37.2i)T2 1 + (-2.32 - 4.03i)T + (-21.5 + 37.2i)T^{2}
47 1+(−6.49−11.2i)T+(−23.5+40.7i)T2 1 + (-6.49 - 11.2i)T + (-23.5 + 40.7i)T^{2}
53 1−1.88T+53T2 1 - 1.88T + 53T^{2}
59 1+(−7.14+12.3i)T+(−29.5−51.0i)T2 1 + (-7.14 + 12.3i)T + (-29.5 - 51.0i)T^{2}
61 1+(−7.15−12.3i)T+(−30.5+52.8i)T2 1 + (-7.15 - 12.3i)T + (-30.5 + 52.8i)T^{2}
67 1+(3.99−6.91i)T+(−33.5−58.0i)T2 1 + (3.99 - 6.91i)T + (-33.5 - 58.0i)T^{2}
71 1+10.2T+71T2 1 + 10.2T + 71T^{2}
73 1+4.98T+73T2 1 + 4.98T + 73T^{2}
79 1+(−4.60−7.97i)T+(−39.5+68.4i)T2 1 + (-4.60 - 7.97i)T + (-39.5 + 68.4i)T^{2}
83 1+(−4.40−7.63i)T+(−41.5+71.8i)T2 1 + (-4.40 - 7.63i)T + (-41.5 + 71.8i)T^{2}
89 1+9.65T+89T2 1 + 9.65T + 89T^{2}
97 1+(−4.32−7.48i)T+(−48.5+84.0i)T2 1 + (-4.32 - 7.48i)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

−9.628862336306115823873257152184, −8.507963438155097461852983914750, −7.61477381262671285275471645510, −7.10980379452563710484960839287, −6.16805722380876958658588015082, −5.55580178916776948421425464961, −4.59111320660843624605352238204, −3.54131843844701919339776408986, −2.47585520508381724223350860315, −0.979330793066210489694237825233, 0.46937208633514497947741358456, 1.82660533984197579320146263773, 3.42032787559572487649908137769, 4.39442907023558392510257909643, 5.04930718727211377081973311919, 5.71327199690212884772058246978, 6.91642658223995165976354794515, 7.30008996912562706906628100996, 8.581642160168505375243193099629, 9.082878253492080175524118103767

Graph of the ZZ-function along the critical line