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Properties

Label 2-1408-176.131-c1-0-0
Degree 22
Conductor 14081408
Sign −0.178+0.983i-0.178 + 0.983i
Analytic cond. 11.242911.2429
Root an. cond. 3.353043.35304
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  + (−1.33 + 1.33i)3-s + (−1.27 + 1.27i)5-s − 0.148i·7-s − 0.586i·9-s + (−3.12 + 1.11i)11-s + (0.488 − 0.488i)13-s − 3.42i·15-s + 3.29i·17-s + (1.88 + 1.88i)19-s + (0.198 + 0.198i)21-s − 7.21·23-s + 1.72i·25-s + (−3.23 − 3.23i)27-s + (4.17 − 4.17i)29-s − 2.16i·31-s + ⋯
L(s)  = 1  + (−0.773 + 0.773i)3-s + (−0.571 + 0.571i)5-s − 0.0560i·7-s − 0.195i·9-s + (−0.942 + 0.335i)11-s + (0.135 − 0.135i)13-s − 0.884i·15-s + 0.799i·17-s + (0.432 + 0.432i)19-s + (0.0433 + 0.0433i)21-s − 1.50·23-s + 0.345i·25-s + (−0.621 − 0.621i)27-s + (0.774 − 0.774i)29-s − 0.388i·31-s + ⋯

Functional equation

Λ(s)=(1408s/2 ΓC(s) L(s)=((−0.178+0.983i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1408s/2 ΓC(s+1/2) L(s)=((−0.178+0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14081408    =    27⋅112^{7} \cdot 11
Sign: −0.178+0.983i-0.178 + 0.983i
Analytic conductor: 11.242911.2429
Root analytic conductor: 3.353043.35304
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1408(351,⋅)\chi_{1408} (351, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1408, ( :1/2), −0.178+0.983i)(2,\ 1408,\ (\ :1/2),\ -0.178 + 0.983i)

Particular Values

L(1)L(1) ≈\approx 0.021468495160.02146849516
L(12)L(\frac12) ≈\approx 0.021468495160.02146849516
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
11 1+(3.12−1.11i)T 1 + (3.12 - 1.11i)T
good3 1+(1.33−1.33i)T−3iT2 1 + (1.33 - 1.33i)T - 3iT^{2}
5 1+(1.27−1.27i)T−5iT2 1 + (1.27 - 1.27i)T - 5iT^{2}
7 1+0.148iT−7T2 1 + 0.148iT - 7T^{2}
13 1+(−0.488+0.488i)T−13iT2 1 + (-0.488 + 0.488i)T - 13iT^{2}
17 1−3.29iT−17T2 1 - 3.29iT - 17T^{2}
19 1+(−1.88−1.88i)T+19iT2 1 + (-1.88 - 1.88i)T + 19iT^{2}
23 1+7.21T+23T2 1 + 7.21T + 23T^{2}
29 1+(−4.17+4.17i)T−29iT2 1 + (-4.17 + 4.17i)T - 29iT^{2}
31 1+2.16iT−31T2 1 + 2.16iT - 31T^{2}
37 1+(−0.771+0.771i)T−37iT2 1 + (-0.771 + 0.771i)T - 37iT^{2}
41 1+8.26T+41T2 1 + 8.26T + 41T^{2}
43 1+(7.14−7.14i)T−43iT2 1 + (7.14 - 7.14i)T - 43iT^{2}
47 1+7.43iT−47T2 1 + 7.43iT - 47T^{2}
53 1+(−6.68+6.68i)T−53iT2 1 + (-6.68 + 6.68i)T - 53iT^{2}
59 1+(1.21+1.21i)T+59iT2 1 + (1.21 + 1.21i)T + 59iT^{2}
61 1+(−2.33+2.33i)T−61iT2 1 + (-2.33 + 2.33i)T - 61iT^{2}
67 1+(−7.02+7.02i)T−67iT2 1 + (-7.02 + 7.02i)T - 67iT^{2}
71 1+4.85T+71T2 1 + 4.85T + 71T^{2}
73 1−8.13T+73T2 1 - 8.13T + 73T^{2}
79 1−5.02T+79T2 1 - 5.02T + 79T^{2}
83 1+(7.65+7.65i)T+83iT2 1 + (7.65 + 7.65i)T + 83iT^{2}
89 1+6.12iT−89T2 1 + 6.12iT - 89T^{2}
97 1+6.62T+97T2 1 + 6.62T + 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

−10.19128275211113603753492189228, −9.770954968479088899650618116106, −8.255458586481483379757562161524, −7.891199379950983200113678306657, −6.81411274414242404050724333401, −5.87608266946666595655456185380, −5.16554352667636829300407443734, −4.21033581999076345586773618602, −3.44596791875234106070090910856, −2.09890041447874685507481577890, 0.01087185747021606746129118818, 1.10976673836411366390108020645, 2.58739565083842242061514038924, 3.83934920186630218251813415242, 4.97759265379992201815245873910, 5.59021414386967013732753118796, 6.58842619117412402916752013893, 7.27868253087633338486187332940, 8.136105224323632860389775782845, 8.766541627914399359201129002377

Graph of the ZZ-function along the critical line