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Properties

Label 2-42e2-9.7-c1-0-14
Degree 22
Conductor 17641764
Sign 0.552−0.833i0.552 - 0.833i
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  + (1.69 + 0.353i)3-s + (−0.469 − 0.812i)5-s + (2.75 + 1.19i)9-s + (−1.31 + 2.28i)11-s + (2.71 + 4.69i)13-s + (−0.508 − 1.54i)15-s − 3.85·17-s + 1.09·19-s + (3.16 + 5.48i)23-s + (2.05 − 3.56i)25-s + (4.24 + 3.00i)27-s + (−1.94 + 3.36i)29-s + (−2.33 − 4.04i)31-s + (−3.04 + 3.40i)33-s + 2.30·37-s + ⋯
L(s)  = 1  + (0.979 + 0.203i)3-s + (−0.209 − 0.363i)5-s + (0.916 + 0.399i)9-s + (−0.397 + 0.688i)11-s + (0.751 + 1.30i)13-s + (−0.131 − 0.398i)15-s − 0.934·17-s + 0.251·19-s + (0.660 + 1.14i)23-s + (0.411 − 0.713i)25-s + (0.816 + 0.577i)27-s + (−0.360 + 0.624i)29-s + (−0.419 − 0.725i)31-s + (−0.529 + 0.592i)33-s + 0.379·37-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((0.552−0.833i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s+1/2) L(s)=((0.552−0.833i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.552−0.833i0.552 - 0.833i
Analytic conductor: 14.085614.0856
Root analytic conductor: 3.753083.75308
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1764(1177,⋅)\chi_{1764} (1177, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :1/2), 0.552−0.833i)(2,\ 1764,\ (\ :1/2),\ 0.552 - 0.833i)

Particular Values

L(1)L(1) ≈\approx 2.3231827242.323182724
L(12)L(\frac12) ≈\approx 2.3231827242.323182724
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+(−1.69−0.353i)T 1 + (-1.69 - 0.353i)T
7 1 1
good5 1+(0.469+0.812i)T+(−2.5+4.33i)T2 1 + (0.469 + 0.812i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.31−2.28i)T+(−5.5−9.52i)T2 1 + (1.31 - 2.28i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.71−4.69i)T+(−6.5+11.2i)T2 1 + (-2.71 - 4.69i)T + (-6.5 + 11.2i)T^{2}
17 1+3.85T+17T2 1 + 3.85T + 17T^{2}
19 1−1.09T+19T2 1 - 1.09T + 19T^{2}
23 1+(−3.16−5.48i)T+(−11.5+19.9i)T2 1 + (-3.16 - 5.48i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.94−3.36i)T+(−14.5−25.1i)T2 1 + (1.94 - 3.36i)T + (-14.5 - 25.1i)T^{2}
31 1+(2.33+4.04i)T+(−15.5+26.8i)T2 1 + (2.33 + 4.04i)T + (-15.5 + 26.8i)T^{2}
37 1−2.30T+37T2 1 - 2.30T + 37T^{2}
41 1+(−4.12−7.13i)T+(−20.5+35.5i)T2 1 + (-4.12 - 7.13i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.14−3.71i)T+(−21.5−37.2i)T2 1 + (2.14 - 3.71i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.32−2.28i)T+(−23.5−40.7i)T2 1 + (1.32 - 2.28i)T + (-23.5 - 40.7i)T^{2}
53 1+1.27T+53T2 1 + 1.27T + 53T^{2}
59 1+(3.02+5.23i)T+(−29.5+51.0i)T2 1 + (3.02 + 5.23i)T + (-29.5 + 51.0i)T^{2}
61 1+(−6.71+11.6i)T+(−30.5−52.8i)T2 1 + (-6.71 + 11.6i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.64−6.31i)T+(−33.5+58.0i)T2 1 + (-3.64 - 6.31i)T + (-33.5 + 58.0i)T^{2}
71 1−14.7T+71T2 1 - 14.7T + 71T^{2}
73 1−5.74T+73T2 1 - 5.74T + 73T^{2}
79 1+(5.51−9.54i)T+(−39.5−68.4i)T2 1 + (5.51 - 9.54i)T + (-39.5 - 68.4i)T^{2}
83 1+(−1.24+2.15i)T+(−41.5−71.8i)T2 1 + (-1.24 + 2.15i)T + (-41.5 - 71.8i)T^{2}
89 1+13.5T+89T2 1 + 13.5T + 89T^{2}
97 1+(1.75−3.03i)T+(−48.5−84.0i)T2 1 + (1.75 - 3.03i)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

−9.427779060536841798490732325669, −8.669649313884591369802831103757, −7.980701999388014397031085282126, −7.14708933107019091876372777572, −6.43536191947359168220630231036, −5.05671385885169713745553396068, −4.37649119714808898729819788378, −3.59100988717328028539353230476, −2.42465255342961919209389013634, −1.47259563181071873567554065750, 0.813487008547812658610409056102, 2.33843160984833997121287921286, 3.15728098356505063822134591280, 3.86061546320768506255029736645, 5.06106711036829587451107764582, 6.05885476538678679749706450730, 6.97823145773251972855698808731, 7.64679560812526509623581273071, 8.558838601422115407081176858887, 8.818785013937507570989210043563

Graph of the ZZ-function along the critical line