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Properties

Label 2-40e2-40.29-c1-0-31
Degree 22
Conductor 16001600
Sign 0.316+0.948i0.316 + 0.948i
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  + 2·3-s + 9-s − 6i·11-s − 6i·17-s − 2i·19-s − 4·27-s − 12i·33-s − 6·41-s + 10·43-s + 7·49-s − 12i·51-s − 4i·57-s + 6i·59-s + 14·67-s − 2i·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.333·9-s − 1.80i·11-s − 1.45i·17-s − 0.458i·19-s − 0.769·27-s − 2.08i·33-s − 0.937·41-s + 1.52·43-s + 49-s − 1.68i·51-s − 0.529i·57-s + 0.781i·59-s + 1.71·67-s − 0.234i·73-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((0.316+0.948i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=((0.316+0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.316+0.948i0.316 + 0.948i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(1249,⋅)\chi_{1600} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), 0.316+0.948i)(2,\ 1600,\ (\ :1/2),\ 0.316 + 0.948i)

Particular Values

L(1)L(1) ≈\approx 2.2529968422.252996842
L(12)L(\frac12) ≈\approx 2.2529968422.252996842
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 1−2T+3T2 1 - 2T + 3T^{2}
7 1−7T2 1 - 7T^{2}
11 1+6iT−11T2 1 + 6iT - 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+6iT−17T2 1 + 6iT - 17T^{2}
19 1+2iT−19T2 1 + 2iT - 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−10T+43T2 1 - 10T + 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1−6iT−59T2 1 - 6iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1−14T+67T2 1 - 14T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+2iT−73T2 1 + 2iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−18T+83T2 1 - 18T + 83T^{2}
89 1+18T+89T2 1 + 18T + 89T^{2}
97 1−10iT−97T2 1 - 10iT - 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

−9.043286683129683219459596029754, −8.591579832571673505528003084598, −7.82336287633584810161167813566, −7.02556548992560164162144608904, −5.96858853173716671788493034377, −5.14925602938534690765238908263, −3.89753377453048086960262528029, −3.08735359786260095590437152702, −2.43016051313926583327314117215, −0.74244896642002666158509946344, 1.71493873068809262695085591796, 2.42481699904604462119437761230, 3.64768968698289207823728794986, 4.29569294913590630432367766231, 5.41852473123892389915316951896, 6.49255958603362394750607504820, 7.38715341089691132133790223606, 8.019838620753455863560859146050, 8.730054488693157569360021493151, 9.534829036418252700257629357443

Graph of the ZZ-function along the critical line