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Properties

Label 2-1445-5.4-c1-0-53
Degree 22
Conductor 14451445
Sign 0.0796−0.996i0.0796 - 0.996i
Analytic cond. 11.538311.5383
Root an. cond. 3.396813.39681
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  + 1.80i·2-s − 0.561i·3-s − 1.25·4-s + (0.178 − 2.22i)5-s + 1.01·6-s + 3.11i·7-s + 1.34i·8-s + 2.68·9-s + (4.01 + 0.321i)10-s + 5.61·11-s + 0.703i·12-s − 1.24i·13-s − 5.61·14-s + (−1.25 − 0.100i)15-s − 4.93·16-s + ⋯
L(s)  = 1  + 1.27i·2-s − 0.324i·3-s − 0.626·4-s + (0.0796 − 0.996i)5-s + 0.413·6-s + 1.17i·7-s + 0.476i·8-s + 0.894·9-s + (1.27 + 0.101i)10-s + 1.69·11-s + 0.203i·12-s − 0.346i·13-s − 1.49·14-s + (−0.323 − 0.0258i)15-s − 1.23·16-s + ⋯

Functional equation

Λ(s)=(1445s/2 ΓC(s) L(s)=((0.0796−0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1445s/2 ΓC(s+1/2) L(s)=((0.0796−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14451445    =    5⋅1725 \cdot 17^{2}
Sign: 0.0796−0.996i0.0796 - 0.996i
Analytic conductor: 11.538311.5383
Root analytic conductor: 3.396813.39681
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1445(579,⋅)\chi_{1445} (579, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1445, ( :1/2), 0.0796−0.996i)(2,\ 1445,\ (\ :1/2),\ 0.0796 - 0.996i)

Particular Values

L(1)L(1) ≈\approx 2.0941216642.094121664
L(12)L(\frac12) ≈\approx 2.0941216642.094121664
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)
bad5 1+(−0.178+2.22i)T 1 + (-0.178 + 2.22i)T
17 1 1
good2 1−1.80iT−2T2 1 - 1.80iT - 2T^{2}
3 1+0.561iT−3T2 1 + 0.561iT - 3T^{2}
7 1−3.11iT−7T2 1 - 3.11iT - 7T^{2}
11 1−5.61T+11T2 1 - 5.61T + 11T^{2}
13 1+1.24iT−13T2 1 + 1.24iT - 13T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+2.32iT−23T2 1 + 2.32iT - 23T^{2}
29 1−6.62T+29T2 1 - 6.62T + 29T^{2}
31 1−4.55T+31T2 1 - 4.55T + 31T^{2}
37 1−1.90iT−37T2 1 - 1.90iT - 37T^{2}
41 1−5.92T+41T2 1 - 5.92T + 41T^{2}
43 1+2.04iT−43T2 1 + 2.04iT - 43T^{2}
47 1−4.85iT−47T2 1 - 4.85iT - 47T^{2}
53 1−9.11iT−53T2 1 - 9.11iT - 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1−5.65T+61T2 1 - 5.65T + 61T^{2}
67 1−8.46iT−67T2 1 - 8.46iT - 67T^{2}
71 1+8.79T+71T2 1 + 8.79T + 71T^{2}
73 1−1.56iT−73T2 1 - 1.56iT - 73T^{2}
79 1+4.91T+79T2 1 + 4.91T + 79T^{2}
83 1+3.94iT−83T2 1 + 3.94iT - 83T^{2}
89 1−10.6T+89T2 1 - 10.6T + 89T^{2}
97 1+12.3iT−97T2 1 + 12.3iT - 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

−9.212628131510390724761480617492, −8.765051028346086558269146991044, −8.154835812460641944604232770760, −7.20501741500977452942458377601, −6.26248373426958390345212444944, −6.01756025923642748296664795802, −4.79252596947461487530513698523, −4.24079418450321334765332286245, −2.43116980635679018404134992694, −1.25071451876621200374146399375, 1.04940620802546342574655172915, 2.02930362778332854636379122518, 3.33378409688452242478373795601, 4.00331426845810679423517450880, 4.47716269252410670676391617901, 6.47033511266062858216457511780, 6.71148967542710863736343278666, 7.58741347106718714168213786347, 8.962169286096430080025993633124, 9.756419001771142555887772462354

Graph of the ZZ-function along the critical line