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Properties

Label 2-40e2-5.4-c1-0-7
Degree 22
Conductor 16001600
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
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  + 3·9-s + 6i·13-s + 2i·17-s − 10·29-s + 2i·37-s + 10·41-s + 7·49-s + 14i·53-s + 10·61-s + 6i·73-s + 9·81-s − 10·89-s + 18i·97-s + 2·101-s + 6·109-s + ⋯
L(s)  = 1  + 9-s + 1.66i·13-s + 0.485i·17-s − 1.85·29-s + 0.328i·37-s + 1.56·41-s + 49-s + 1.92i·53-s + 1.28·61-s + 0.702i·73-s + 81-s − 1.05·89-s + 1.82i·97-s + 0.199·101-s + 0.574·109-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(449,⋅)\chi_{1600} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), 0.447−0.894i)(2,\ 1600,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.6583348051.658334805
L(12)L(\frac12) ≈\approx 1.6583348051.658334805
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 1
good3 1−3T2 1 - 3T^{2}
7 1−7T2 1 - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−6iT−13T2 1 - 6iT - 13T^{2}
17 1−2iT−17T2 1 - 2iT - 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1+10T+29T2 1 + 10T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1−10T+41T2 1 - 10T + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1−14iT−53T2 1 - 14iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−6iT−73T2 1 - 6iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1−18iT−97T2 1 - 18iT - 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

−9.379985788756507890248421986987, −9.016694792407669942144884550905, −7.81098404526705947035644607784, −7.16211540055356908696640317727, −6.42305597257860151217384376818, −5.47609993860249359551185515327, −4.29583604653231160094339269388, −3.90274935645140785862603538659, −2.34529288830870757534092297515, −1.37774851511004780680395899227, 0.68967941519025122210458517661, 2.09752032955479824021626459538, 3.28514445811830652857474085937, 4.16525659046008017832538891650, 5.24094691624264550546619660031, 5.87475744794244571478260991634, 7.07700643386195257397833728416, 7.57493599446690188721245822483, 8.400361364554562061744997916979, 9.421389577647872954165604973444

Graph of the ZZ-function along the critical line