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Properties

Label 2-40e2-5.4-c1-0-4
Degree 22
Conductor 16001600
Sign 0.894−0.447i0.894 - 0.447i
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  − 3i·3-s + 2i·7-s − 6·9-s − 11-s + 4i·13-s + 5i·17-s + 19-s + 6·21-s + 2i·23-s + 9i·27-s − 8·29-s + 10·31-s + 3i·33-s + 6i·37-s + 12·39-s + ⋯
L(s)  = 1  − 1.73i·3-s + 0.755i·7-s − 2·9-s − 0.301·11-s + 1.10i·13-s + 1.21i·17-s + 0.229·19-s + 1.30·21-s + 0.417i·23-s + 1.73i·27-s − 1.48·29-s + 1.79·31-s + 0.522i·33-s + 0.986i·37-s + 1.92·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.894−0.447i0.894 - 0.447i
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), 0.894−0.447i)(2,\ 1600,\ (\ :1/2),\ 0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.1310974381.131097438
L(12)L(\frac12) ≈\approx 1.1310974381.131097438
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+3iT−3T2 1 + 3iT - 3T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+T+11T2 1 + T + 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1−5iT−17T2 1 - 5iT - 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1−2iT−23T2 1 - 2iT - 23T^{2}
29 1+8T+29T2 1 + 8T + 29T^{2}
31 1−10T+31T2 1 - 10T + 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1+3T+41T2 1 + 3T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−4iT−47T2 1 - 4iT - 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 1−iT−67T2 1 - iT - 67T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 1+3iT−73T2 1 + 3iT - 73T^{2}
79 1+6T+79T2 1 + 6T + 79T^{2}
83 1+13iT−83T2 1 + 13iT - 83T^{2}
89 1−9T+89T2 1 - 9T + 89T^{2}
97 1+14iT−97T2 1 + 14iT - 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

−9.172233008589101017225964234474, −8.513325630923809575768664163890, −7.81236172682069523212773415653, −7.09745507008846527341507454473, −6.20563951633160571426882134350, −5.82959345824437763961307185915, −4.54389805745358666724377655179, −3.12842516438000579818624841082, −2.12685928326681266783326889312, −1.36799207208325963679370961364, 0.44636699284835547915522807559, 2.66411158964718331863472444671, 3.50108155382465016065228418309, 4.30551214951101528837659676053, 5.12463359720506145016660818974, 5.70906291585543162607006596579, 7.01624557358276738604468088384, 7.894062691419872661851029667603, 8.746598502943036555869545077196, 9.558458105415433116124152083751

Graph of the ZZ-function along the critical line