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Properties

Label 2-2e7-8.5-c3-0-1
Degree 22
Conductor 128128
Sign −i-i
Analytic cond. 7.552247.55224
Root an. cond. 2.748132.74813
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·3-s + 19·9-s + 70.7i·11-s − 90·17-s + 127. i·19-s + 125·25-s + 130. i·27-s − 200.·33-s + 522·41-s − 483. i·43-s − 343·49-s − 254. i·51-s − 360.·57-s − 325. i·59-s − 1.09e3i·67-s + ⋯
L(s)  = 1  + 0.544i·3-s + 0.703·9-s + 1.93i·11-s − 1.28·17-s + 1.53i·19-s + 25-s + 0.927i·27-s − 1.05·33-s + 1.98·41-s − 1.71i·43-s − 49-s − 0.698i·51-s − 0.836·57-s − 0.717i·59-s − 1.99i·67-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=(−i Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+3/2) L(s)=(−i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: −i-i
Analytic conductor: 7.552247.55224
Root analytic conductor: 2.748132.74813
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ128(65,⋅)\chi_{128} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 128, ( :3/2), −i)(2,\ 128,\ (\ :3/2),\ -i)

Particular Values

L(2)L(2) ≈\approx 1.06949+1.06949i1.06949 + 1.06949i
L(12)L(\frac12) ≈\approx 1.06949+1.06949i1.06949 + 1.06949i
L(52)L(\frac{5}{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
good3 1−2.82iT−27T2 1 - 2.82iT - 27T^{2}
5 1−125T2 1 - 125T^{2}
7 1+343T2 1 + 343T^{2}
11 1−70.7iT−1.33e3T2 1 - 70.7iT - 1.33e3T^{2}
13 1−2.19e3T2 1 - 2.19e3T^{2}
17 1+90T+4.91e3T2 1 + 90T + 4.91e3T^{2}
19 1−127.iT−6.85e3T2 1 - 127. iT - 6.85e3T^{2}
23 1+1.21e4T2 1 + 1.21e4T^{2}
29 1−2.43e4T2 1 - 2.43e4T^{2}
31 1+2.97e4T2 1 + 2.97e4T^{2}
37 1−5.06e4T2 1 - 5.06e4T^{2}
41 1−522T+6.89e4T2 1 - 522T + 6.89e4T^{2}
43 1+483.iT−7.95e4T2 1 + 483. iT - 7.95e4T^{2}
47 1+1.03e5T2 1 + 1.03e5T^{2}
53 1−1.48e5T2 1 - 1.48e5T^{2}
59 1+325.iT−2.05e5T2 1 + 325. iT - 2.05e5T^{2}
61 1−2.26e5T2 1 - 2.26e5T^{2}
67 1+1.09e3iT−3.00e5T2 1 + 1.09e3iT - 3.00e5T^{2}
71 1+3.57e5T2 1 + 3.57e5T^{2}
73 1−430T+3.89e5T2 1 - 430T + 3.89e5T^{2}
79 1+4.93e5T2 1 + 4.93e5T^{2}
83 1−681.iT−5.71e5T2 1 - 681. iT - 5.71e5T^{2}
89 1+1.02e3T+7.04e5T2 1 + 1.02e3T + 7.04e5T^{2}
97 1−1.91e3T+9.12e5T2 1 - 1.91e3T + 9.12e5T^{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

−12.86613391569272739481506734153, −12.32145352280495509802959818033, −10.84174934320969070722998291614, −9.996913186961124265760855940422, −9.162901398394641071451612150416, −7.65244323634178350286896130015, −6.64124692147669390435640856039, −4.91964601291452429483575426540, −4.01134026870600113444691267406, −1.94800741825499369940914819677, 0.819017669647580252860004126787, 2.79117855394793387110811803056, 4.50670355090226620700501858155, 6.10170787573384112802259291180, 7.05336301730707782102125372761, 8.359883756521301568888731607607, 9.267765683929770566087148110692, 10.83996791860317420113309508985, 11.42036153002511934762883501937, 12.94212640267255725999332712455

Graph of the ZZ-function along the critical line