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Properties

Label 2-528-33.32-c3-0-36
Degree 22
Conductor 528528
Sign 0.677+0.735i0.677 + 0.735i
Analytic cond. 31.153031.1530
Root an. cond. 5.581485.58148
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  + (−0.832 − 5.12i)3-s + 6.09i·5-s + 1.16i·7-s + (−25.6 + 8.54i)9-s + (30.4 − 20.0i)11-s + 37.5i·13-s + (31.2 − 5.07i)15-s − 40.8·17-s − 84.3i·19-s + (6 − 0.973i)21-s + 101. i·23-s + 87.7·25-s + (65.1 + 124. i)27-s + 251.·29-s − 19.9·31-s + ⋯
L(s)  = 1  + (−0.160 − 0.987i)3-s + 0.545i·5-s + 0.0631i·7-s + (−0.948 + 0.316i)9-s + (0.834 − 0.550i)11-s + 0.801i·13-s + (0.538 − 0.0874i)15-s − 0.583·17-s − 1.01i·19-s + (0.0623 − 0.0101i)21-s + 0.917i·23-s + 0.702·25-s + (0.464 + 0.885i)27-s + 1.61·29-s − 0.115·31-s + ⋯

Functional equation

Λ(s)=(528s/2 ΓC(s) L(s)=((0.677+0.735i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(528s/2 ΓC(s+3/2) L(s)=((0.677+0.735i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 528528    =    24⋅3⋅112^{4} \cdot 3 \cdot 11
Sign: 0.677+0.735i0.677 + 0.735i
Analytic conductor: 31.153031.1530
Root analytic conductor: 5.581485.58148
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ528(65,⋅)\chi_{528} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 528, ( :3/2), 0.677+0.735i)(2,\ 528,\ (\ :3/2),\ 0.677 + 0.735i)

Particular Values

L(2)L(2) ≈\approx 1.7953035791.795303579
L(12)L(\frac12) ≈\approx 1.7953035791.795303579
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
3 1+(0.832+5.12i)T 1 + (0.832 + 5.12i)T
11 1+(−30.4+20.0i)T 1 + (-30.4 + 20.0i)T
good5 1−6.09iT−125T2 1 - 6.09iT - 125T^{2}
7 1−1.16iT−343T2 1 - 1.16iT - 343T^{2}
13 1−37.5iT−2.19e3T2 1 - 37.5iT - 2.19e3T^{2}
17 1+40.8T+4.91e3T2 1 + 40.8T + 4.91e3T^{2}
19 1+84.3iT−6.85e3T2 1 + 84.3iT - 6.85e3T^{2}
23 1−101.iT−1.21e4T2 1 - 101. iT - 1.21e4T^{2}
29 1−251.T+2.43e4T2 1 - 251.T + 2.43e4T^{2}
31 1+19.9T+2.97e4T2 1 + 19.9T + 2.97e4T^{2}
37 1−45.9T+5.06e4T2 1 - 45.9T + 5.06e4T^{2}
41 1−175.T+6.89e4T2 1 - 175.T + 6.89e4T^{2}
43 1+332.iT−7.95e4T2 1 + 332. iT - 7.95e4T^{2}
47 1+186.iT−1.03e5T2 1 + 186. iT - 1.03e5T^{2}
53 1−554.iT−1.48e5T2 1 - 554. iT - 1.48e5T^{2}
59 1+185.iT−2.05e5T2 1 + 185. iT - 2.05e5T^{2}
61 1+754.iT−2.26e5T2 1 + 754. iT - 2.26e5T^{2}
67 1−381.T+3.00e5T2 1 - 381.T + 3.00e5T^{2}
71 1−401.iT−3.57e5T2 1 - 401. iT - 3.57e5T^{2}
73 1+1.03e3iT−3.89e5T2 1 + 1.03e3iT - 3.89e5T^{2}
79 1+1.04e3iT−4.93e5T2 1 + 1.04e3iT - 4.93e5T^{2}
83 1+847.T+5.71e5T2 1 + 847.T + 5.71e5T^{2}
89 1+675.iT−7.04e5T2 1 + 675. iT - 7.04e5T^{2}
97 1+378.T+9.12e5T2 1 + 378.T + 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

−10.57930007382042523473655961129, −9.170793628332851044220494348248, −8.624110542028990867652923282170, −7.35300044085221125961225324040, −6.72741743537265833318179175083, −6.01042737278408815456783264628, −4.68318645704223091928938680589, −3.25435489649875626818945989979, −2.12554722748825734783020924070, −0.77430074811755968283667402670, 0.922404366223222147684099599659, 2.73354495428806864837331714260, 4.05587411430549938981826959301, 4.72592521993938282703751828077, 5.79640267622573867040098299926, 6.76294970263568932509224677881, 8.197781752169603941239558676887, 8.816369777338215693822202499966, 9.790640729555926137096261835791, 10.40247421478263601357046913459

Graph of the ZZ-function along the critical line