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Properties

Label 2-3168-88.43-c1-0-21
Degree 22
Conductor 31683168
Sign 0.884−0.466i0.884 - 0.466i
Analytic cond. 25.296625.2966
Root an. cond. 5.029575.02957
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  − 2i·5-s − 0.936·7-s + (3.09 + 1.19i)11-s + 4.27·13-s + 3.33i·17-s + 2.89i·19-s + 5.12i·23-s + 25-s − 6.60·29-s + 1.73i·31-s + 1.87i·35-s − 6.18i·37-s + 1.46i·41-s + 10.3i·43-s + 6i·47-s + ⋯
L(s)  = 1  − 0.894i·5-s − 0.353·7-s + (0.932 + 0.361i)11-s + 1.18·13-s + 0.808i·17-s + 0.664i·19-s + 1.06i·23-s + 0.200·25-s − 1.22·29-s + 0.311i·31-s + 0.316i·35-s − 1.01i·37-s + 0.228i·41-s + 1.57i·43-s + 0.875i·47-s + ⋯

Functional equation

Λ(s)=(3168s/2 ΓC(s) L(s)=((0.884−0.466i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3168s/2 ΓC(s+1/2) L(s)=((0.884−0.466i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31683168    =    25⋅32⋅112^{5} \cdot 3^{2} \cdot 11
Sign: 0.884−0.466i0.884 - 0.466i
Analytic conductor: 25.296625.2966
Root analytic conductor: 5.029575.02957
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3168(2287,⋅)\chi_{3168} (2287, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3168, ( :1/2), 0.884−0.466i)(2,\ 3168,\ (\ :1/2),\ 0.884 - 0.466i)

Particular Values

L(1)L(1) ≈\approx 1.8508675091.850867509
L(12)L(\frac12) ≈\approx 1.8508675091.850867509
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
3 1 1
11 1+(−3.09−1.19i)T 1 + (-3.09 - 1.19i)T
good5 1+2iT−5T2 1 + 2iT - 5T^{2}
7 1+0.936T+7T2 1 + 0.936T + 7T^{2}
13 1−4.27T+13T2 1 - 4.27T + 13T^{2}
17 1−3.33iT−17T2 1 - 3.33iT - 17T^{2}
19 1−2.89iT−19T2 1 - 2.89iT - 19T^{2}
23 1−5.12iT−23T2 1 - 5.12iT - 23T^{2}
29 1+6.60T+29T2 1 + 6.60T + 29T^{2}
31 1−1.73iT−31T2 1 - 1.73iT - 31T^{2}
37 1+6.18iT−37T2 1 + 6.18iT - 37T^{2}
41 1−1.46iT−41T2 1 - 1.46iT - 41T^{2}
43 1−10.3iT−43T2 1 - 10.3iT - 43T^{2}
47 1−6iT−47T2 1 - 6iT - 47T^{2}
53 1+8.24iT−53T2 1 + 8.24iT - 53T^{2}
59 1+9.65T+59T2 1 + 9.65T + 59T^{2}
61 1−9.06T+61T2 1 - 9.06T + 61T^{2}
67 1−10.2T+67T2 1 - 10.2T + 67T^{2}
71 1−4.24iT−71T2 1 - 4.24iT - 71T^{2}
73 1−13.2iT−73T2 1 - 13.2iT - 73T^{2}
79 1−3.86T+79T2 1 - 3.86T + 79T^{2}
83 1+2.39iT−83T2 1 + 2.39iT - 83T^{2}
89 1−3.47T+89T2 1 - 3.47T + 89T^{2}
97 1−11.3T+97T2 1 - 11.3T + 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

−8.720909249270766169884917166140, −8.148694924363536139914302647148, −7.27200621499959668030634165970, −6.31087408422923864273019376771, −5.82116734962437204090324339267, −4.86694165575632560531188715895, −3.91389975816825625013739133460, −3.44771172225608755101100585235, −1.82385012106730570000373933504, −1.12217816829237669145323765787, 0.66663176879635249029339811664, 2.05909616162718705863261345192, 3.14399966658414402496515791549, 3.68046069833180237839931684551, 4.70288815138285138962597528707, 5.76425043798023047718191896170, 6.55234659380919833417416966716, 6.86673708704444352283966580699, 7.82129313328336558699543828782, 8.814519771811710024311911280429

Graph of the ZZ-function along the critical line