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Properties

Label 2-7488-12.11-c1-0-13
Degree 22
Conductor 74887488
Sign −0.816−0.577i-0.816 - 0.577i
Analytic cond. 59.791959.7919
Root an. cond. 7.732527.73252
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·7-s − 1.41·11-s − 13-s + 1.41i·17-s + 4i·19-s + 2.82·23-s + 5·25-s + 9.89i·29-s + 2·37-s − 2.82i·41-s − 4i·43-s − 9.89·47-s + 3·49-s − 4.24i·53-s − 1.41·59-s + ⋯
L(s)  = 1  + 0.755i·7-s − 0.426·11-s − 0.277·13-s + 0.342i·17-s + 0.917i·19-s + 0.589·23-s + 25-s + 1.83i·29-s + 0.328·37-s − 0.441i·41-s − 0.609i·43-s − 1.44·47-s + 0.428·49-s − 0.582i·53-s − 0.184·59-s + ⋯

Functional equation

Λ(s)=(7488s/2 ΓC(s) L(s)=((−0.816−0.577i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(7488s/2 ΓC(s+1/2) L(s)=((−0.816−0.577i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 74887488    =    26⋅32⋅132^{6} \cdot 3^{2} \cdot 13
Sign: −0.816−0.577i-0.816 - 0.577i
Analytic conductor: 59.791959.7919
Root analytic conductor: 7.732527.73252
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ7488(4031,⋅)\chi_{7488} (4031, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 7488, ( :1/2), −0.816−0.577i)(2,\ 7488,\ (\ :1/2),\ -0.816 - 0.577i)

Particular Values

L(1)L(1) ≈\approx 1.0765683951.076568395
L(12)L(\frac12) ≈\approx 1.0765683951.076568395
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
13 1+T 1 + T
good5 1−5T2 1 - 5T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+1.41T+11T2 1 + 1.41T + 11T^{2}
17 1−1.41iT−17T2 1 - 1.41iT - 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1−2.82T+23T2 1 - 2.82T + 23T^{2}
29 1−9.89iT−29T2 1 - 9.89iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+2.82iT−41T2 1 + 2.82iT - 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+9.89T+47T2 1 + 9.89T + 47T^{2}
53 1+4.24iT−53T2 1 + 4.24iT - 53T^{2}
59 1+1.41T+59T2 1 + 1.41T + 59T^{2}
61 1+8T+61T2 1 + 8T + 61T^{2}
67 1+10iT−67T2 1 + 10iT - 67T^{2}
71 1−7.07T+71T2 1 - 7.07T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−8iT−79T2 1 - 8iT - 79T^{2}
83 1+12.7T+83T2 1 + 12.7T + 83T^{2}
89 1+5.65iT−89T2 1 + 5.65iT - 89T^{2}
97 1−2T+97T2 1 - 2T + 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

−8.259653709485818782199536262385, −7.47301798124843684061664161442, −6.77456470613577402512093422331, −6.08283342615326817445650242079, −5.22418583352148922284503006884, −4.90695935661309983518185932572, −3.72084106637633549003127158598, −3.05410932742519117314909649907, −2.19072989636188566162693606694, −1.25214381098525281955166205498, 0.27150572431376904006040150794, 1.24019311921885996675135745622, 2.52205122614482167652039647396, 3.06215217720542047608658397282, 4.18515884012541841582530939555, 4.68579080409422515425999368786, 5.42539961956459633495941206236, 6.36475446483165807490649111671, 6.93541139854434095665543617136, 7.61280283418713411662956862006

Graph of the ZZ-function along the critical line