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Properties

Label 2-2e4-4.3-c4-0-1
Degree 22
Conductor 1616
Sign 0.5+0.866i0.5 + 0.866i
Analytic cond. 1.653911.65391
Root an. cond. 1.286041.28604
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.8i·3-s + 18·5-s + 27.7i·7-s − 110.·9-s + 124. i·11-s + 178·13-s − 249. i·15-s − 126·17-s − 401. i·19-s + 383.·21-s + 748. i·23-s − 301·25-s + 415. i·27-s − 1.42e3·29-s − 332. i·31-s + ⋯
L(s)  = 1  − 1.53i·3-s + 0.719·5-s + 0.565i·7-s − 1.37·9-s + 1.03i·11-s + 1.05·13-s − 1.10i·15-s − 0.435·17-s − 1.11i·19-s + 0.870·21-s + 1.41i·23-s − 0.481·25-s + 0.570i·27-s − 1.69·29-s − 0.346i·31-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=((0.5+0.866i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+2) L(s)=((0.5+0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: 0.5+0.866i0.5 + 0.866i
Analytic conductor: 1.653911.65391
Root analytic conductor: 1.286041.28604
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ16(15,⋅)\chi_{16} (15, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 16, ( :2), 0.5+0.866i)(2,\ 16,\ (\ :2),\ 0.5 + 0.866i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.09266−0.630851i1.09266 - 0.630851i
L(12)L(\frac12) ≈\approx 1.09266−0.630851i1.09266 - 0.630851i
L(3)L(3) 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+13.8iT−81T2 1 + 13.8iT - 81T^{2}
5 1−18T+625T2 1 - 18T + 625T^{2}
7 1−27.7iT−2.40e3T2 1 - 27.7iT - 2.40e3T^{2}
11 1−124.iT−1.46e4T2 1 - 124. iT - 1.46e4T^{2}
13 1−178T+2.85e4T2 1 - 178T + 2.85e4T^{2}
17 1+126T+8.35e4T2 1 + 126T + 8.35e4T^{2}
19 1+401.iT−1.30e5T2 1 + 401. iT - 1.30e5T^{2}
23 1−748.iT−2.79e5T2 1 - 748. iT - 2.79e5T^{2}
29 1+1.42e3T+7.07e5T2 1 + 1.42e3T + 7.07e5T^{2}
31 1+332.iT−9.23e5T2 1 + 332. iT - 9.23e5T^{2}
37 1−530T+1.87e6T2 1 - 530T + 1.87e6T^{2}
41 1−162T+2.82e6T2 1 - 162T + 2.82e6T^{2}
43 1+1.53e3iT−3.41e6T2 1 + 1.53e3iT - 3.41e6T^{2}
47 1−3.49e3iT−4.87e6T2 1 - 3.49e3iT - 4.87e6T^{2}
53 1−594T+7.89e6T2 1 - 594T + 7.89e6T^{2}
59 1+2.36e3iT−1.21e7T2 1 + 2.36e3iT - 1.21e7T^{2}
61 1−626T+1.38e7T2 1 - 626T + 1.38e7T^{2}
67 1−1.09e3iT−2.01e7T2 1 - 1.09e3iT - 2.01e7T^{2}
71 1+7.73e3iT−2.54e7T2 1 + 7.73e3iT - 2.54e7T^{2}
73 1+6.68e3T+2.83e7T2 1 + 6.68e3T + 2.83e7T^{2}
79 1−1.38e3iT−3.89e7T2 1 - 1.38e3iT - 3.89e7T^{2}
83 1+4.61e3iT−4.74e7T2 1 + 4.61e3iT - 4.74e7T^{2}
89 1−8.22e3T+6.27e7T2 1 - 8.22e3T + 6.27e7T^{2}
97 1+1.59e3T+8.85e7T2 1 + 1.59e3T + 8.85e7T^{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

−18.06836023437643585814049186767, −17.44936760526088162295216829859, −15.36696591063106557177972043046, −13.64759522673011492851164922547, −12.89920524090815261321697102276, −11.46607753553729154749631859307, −9.217531861594741111994238977521, −7.41165828246141512397470417158, −5.94309129791928061554367690148, −1.92229369019212028723216941492, 3.86456321740832332010452948237, 5.83299655036539627994944390209, 8.715598346365770058092297180882, 10.12250259294611252202556742692, 11.08207130663272797544744924450, 13.46465136652306531144306691342, 14.67508805471552722410182275946, 16.15714647425047230237525206001, 16.84154656385198925609835861715, 18.49403857721936048837836211054

Graph of the ZZ-function along the critical line