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Properties

Label 2-42e2-7.4-c3-0-12
Degree 22
Conductor 17641764
Sign 0.701−0.712i0.701 - 0.712i
Analytic cond. 104.079104.079
Root an. cond. 10.201910.2019
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3 − 5.19i)5-s + (18 − 31.1i)11-s − 62·13-s + (−57 + 98.7i)17-s + (−38 − 65.8i)19-s + (−12 − 20.7i)23-s + (44.5 − 77.0i)25-s − 54·29-s + (−56 + 96.9i)31-s + (89 + 154. i)37-s + 378·41-s − 172·43-s + (96 + 166. i)47-s + (−201 + 348. i)53-s − 216·55-s + ⋯
L(s)  = 1  + (−0.268 − 0.464i)5-s + (0.493 − 0.854i)11-s − 1.32·13-s + (−0.813 + 1.40i)17-s + (−0.458 − 0.794i)19-s + (−0.108 − 0.188i)23-s + (0.355 − 0.616i)25-s − 0.345·29-s + (−0.324 + 0.561i)31-s + (0.395 + 0.684i)37-s + 1.43·41-s − 0.609·43-s + (0.297 + 0.516i)47-s + (−0.520 + 0.902i)53-s − 0.529·55-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((0.701−0.712i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s+3/2) L(s)=((0.701−0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.701−0.712i0.701 - 0.712i
Analytic conductor: 104.079104.079
Root analytic conductor: 10.201910.2019
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1764(361,⋅)\chi_{1764} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :3/2), 0.701−0.712i)(2,\ 1764,\ (\ :3/2),\ 0.701 - 0.712i)

Particular Values

L(2)L(2) ≈\approx 1.1546583531.154658353
L(12)L(\frac12) ≈\approx 1.1546583531.154658353
L(52)L(\frac{5}{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
7 1 1
good5 1+(3+5.19i)T+(−62.5+108.i)T2 1 + (3 + 5.19i)T + (-62.5 + 108. i)T^{2}
11 1+(−18+31.1i)T+(−665.5−1.15e3i)T2 1 + (-18 + 31.1i)T + (-665.5 - 1.15e3i)T^{2}
13 1+62T+2.19e3T2 1 + 62T + 2.19e3T^{2}
17 1+(57−98.7i)T+(−2.45e3−4.25e3i)T2 1 + (57 - 98.7i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(38+65.8i)T+(−3.42e3+5.94e3i)T2 1 + (38 + 65.8i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(12+20.7i)T+(−6.08e3+1.05e4i)T2 1 + (12 + 20.7i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+54T+2.43e4T2 1 + 54T + 2.43e4T^{2}
31 1+(56−96.9i)T+(−1.48e4−2.57e4i)T2 1 + (56 - 96.9i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(−89−154.i)T+(−2.53e4+4.38e4i)T2 1 + (-89 - 154. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1−378T+6.89e4T2 1 - 378T + 6.89e4T^{2}
43 1+172T+7.95e4T2 1 + 172T + 7.95e4T^{2}
47 1+(−96−166.i)T+(−5.19e4+8.99e4i)T2 1 + (-96 - 166. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(201−348.i)T+(−7.44e4−1.28e5i)T2 1 + (201 - 348. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(198−342.i)T+(−1.02e5−1.77e5i)T2 1 + (198 - 342. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−127−219.i)T+(−1.13e5+1.96e5i)T2 1 + (-127 - 219. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−506+876.i)T+(−1.50e5−2.60e5i)T2 1 + (-506 + 876. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+840T+3.57e5T2 1 + 840T + 3.57e5T^{2}
73 1+(−445+770.i)T+(−1.94e5−3.36e5i)T2 1 + (-445 + 770. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(40+69.2i)T+(−2.46e5+4.26e5i)T2 1 + (40 + 69.2i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+108T+5.71e5T2 1 + 108T + 5.71e5T^{2}
89 1+(−819−1.41e3i)T+(−3.52e5+6.10e5i)T2 1 + (-819 - 1.41e3i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1+1.01e3T+9.12e5T2 1 + 1.01e3T + 9.12e5T^{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

−8.923497652162725536121609070862, −8.377416197079429333854448799420, −7.52185868061613281143560104556, −6.57334375772520202614523992490, −5.91861075097303946200358644003, −4.74411503840821895583870477903, −4.23843192129614131371634627072, −3.06001763876551630952708410185, −2.01840154608359979620468136520, −0.73809328277203389995923891327, 0.33109818455107951230742245447, 1.92024308705085918366006901710, 2.70456784454764596783976509523, 3.88302496441562025885601956134, 4.67221805095848334298301244823, 5.52363707822920626694777538804, 6.69320365203738603565965598293, 7.23050638232465351042308915128, 7.82001890065565460814948374560, 9.029255558592590997349599013697

Graph of the ZZ-function along the critical line