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Properties

Label 2-2240-5.4-c1-0-17
Degree 22
Conductor 22402240
Sign 0.332−0.943i0.332 - 0.943i
Analytic cond. 17.886417.8864
Root an. cond. 4.229244.22924
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  + 0.321i·3-s + (−2.10 − 0.742i)5-s + i·7-s + 2.89·9-s + 4.37·11-s + 5.86i·13-s + (0.238 − 0.678i)15-s − 4.85i·17-s − 7.75·19-s − 0.321·21-s + 1.35i·23-s + (3.89 + 3.13i)25-s + 1.89i·27-s − 0.539·29-s + 2.97·31-s + ⋯
L(s)  = 1  + 0.185i·3-s + (−0.943 − 0.332i)5-s + 0.377i·7-s + 0.965·9-s + 1.32·11-s + 1.62i·13-s + (0.0616 − 0.175i)15-s − 1.17i·17-s − 1.77·19-s − 0.0701·21-s + 0.282i·23-s + (0.779 + 0.626i)25-s + 0.364i·27-s − 0.100·29-s + 0.533·31-s + ⋯

Functional equation

Λ(s)=(2240s/2 ΓC(s) L(s)=((0.332−0.943i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2240s/2 ΓC(s+1/2) L(s)=((0.332−0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22402240    =    26⋅5⋅72^{6} \cdot 5 \cdot 7
Sign: 0.332−0.943i0.332 - 0.943i
Analytic conductor: 17.886417.8864
Root analytic conductor: 4.229244.22924
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2240(449,⋅)\chi_{2240} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2240, ( :1/2), 0.332−0.943i)(2,\ 2240,\ (\ :1/2),\ 0.332 - 0.943i)

Particular Values

L(1)L(1) ≈\approx 1.4651145941.465114594
L(12)L(\frac12) ≈\approx 1.4651145941.465114594
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+(2.10+0.742i)T 1 + (2.10 + 0.742i)T
7 1−iT 1 - iT
good3 1−0.321iT−3T2 1 - 0.321iT - 3T^{2}
11 1−4.37T+11T2 1 - 4.37T + 11T^{2}
13 1−5.86iT−13T2 1 - 5.86iT - 13T^{2}
17 1+4.85iT−17T2 1 + 4.85iT - 17T^{2}
19 1+7.75T+19T2 1 + 7.75T + 19T^{2}
23 1−1.35iT−23T2 1 - 1.35iT - 23T^{2}
29 1+0.539T+29T2 1 + 0.539T + 29T^{2}
31 1−2.97T+31T2 1 - 2.97T + 31T^{2}
37 1+6.26iT−37T2 1 + 6.26iT - 37T^{2}
41 1−2.64T+41T2 1 - 2.64T + 41T^{2}
43 1−4.64iT−43T2 1 - 4.64iT - 43T^{2}
47 1−10.3iT−47T2 1 - 10.3iT - 47T^{2}
53 1+0.477iT−53T2 1 + 0.477iT - 53T^{2}
59 1−7.75T+59T2 1 - 7.75T + 59T^{2}
61 1+7.57T+61T2 1 + 7.57T + 61T^{2}
67 1−3.79iT−67T2 1 - 3.79iT - 67T^{2}
71 1−9.23T+71T2 1 - 9.23T + 71T^{2}
73 1+0.477iT−73T2 1 + 0.477iT - 73T^{2}
79 1−1.88T+79T2 1 - 1.88T + 79T^{2}
83 1−15.2iT−83T2 1 - 15.2iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1−13.6iT−97T2 1 - 13.6iT - 97T^{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

−9.260404950172923991703700960046, −8.568073278959866068350931822771, −7.57990284718845400759662936915, −6.82671387562040942010472874569, −6.32500345932592247120896873676, −4.88360662930132680278869394981, −4.25540321350106664884965938986, −3.80795192161127660813587342976, −2.30808724706677274581712361477, −1.17119617808064601953412422157, 0.59156304385804532084819310439, 1.83553166227328797718539638765, 3.25885984458710767186569355426, 4.02235572933946471316617325942, 4.55456795228821721050777603301, 5.96144058434331793148989215389, 6.68199428368724849918548145299, 7.25369623979183009850269469117, 8.314724761261167874406036377751, 8.451498976419150457144127108582

Graph of the ZZ-function along the critical line