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Properties

Label 2-1620-9.7-c3-0-45
Degree 22
Conductor 16201620
Sign −0.939+0.342i-0.939 + 0.342i
Analytic cond. 95.583095.5830
Root an. cond. 9.776669.77666
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  + (−2.5 − 4.33i)5-s + (14 − 24.2i)7-s + (−12 + 20.7i)11-s + (35 + 60.6i)13-s − 102·17-s + 20·19-s + (−36 − 62.3i)23-s + (−12.5 + 21.6i)25-s + (153 − 265. i)29-s + (68 + 117. i)31-s − 140·35-s − 214·37-s + (−75 − 129. i)41-s + (146 − 252. i)43-s + (−36 + 62.3i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.755 − 1.30i)7-s + (−0.328 + 0.569i)11-s + (0.746 + 1.29i)13-s − 1.45·17-s + 0.241·19-s + (−0.326 − 0.565i)23-s + (−0.100 + 0.173i)25-s + (0.979 − 1.69i)29-s + (0.393 + 0.682i)31-s − 0.676·35-s − 0.950·37-s + (−0.285 − 0.494i)41-s + (0.517 − 0.896i)43-s + (−0.111 + 0.193i)47-s + ⋯

Functional equation

Λ(s)=(1620s/2 ΓC(s) L(s)=((−0.939+0.342i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1620s/2 ΓC(s+3/2) L(s)=((−0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: −0.939+0.342i-0.939 + 0.342i
Analytic conductor: 95.583095.5830
Root analytic conductor: 9.776669.77666
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1620(541,⋅)\chi_{1620} (541, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1620, ( :3/2), −0.939+0.342i)(2,\ 1620,\ (\ :3/2),\ -0.939 + 0.342i)

Particular Values

L(2)L(2) ≈\approx 0.93845476710.9384547671
L(12)L(\frac12) ≈\approx 0.93845476710.9384547671
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 1
5 1+(2.5+4.33i)T 1 + (2.5 + 4.33i)T
good7 1+(−14+24.2i)T+(−171.5−297.i)T2 1 + (-14 + 24.2i)T + (-171.5 - 297. i)T^{2}
11 1+(12−20.7i)T+(−665.5−1.15e3i)T2 1 + (12 - 20.7i)T + (-665.5 - 1.15e3i)T^{2}
13 1+(−35−60.6i)T+(−1.09e3+1.90e3i)T2 1 + (-35 - 60.6i)T + (-1.09e3 + 1.90e3i)T^{2}
17 1+102T+4.91e3T2 1 + 102T + 4.91e3T^{2}
19 1−20T+6.85e3T2 1 - 20T + 6.85e3T^{2}
23 1+(36+62.3i)T+(−6.08e3+1.05e4i)T2 1 + (36 + 62.3i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−153+265.i)T+(−1.21e4−2.11e4i)T2 1 + (-153 + 265. i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+(−68−117.i)T+(−1.48e4+2.57e4i)T2 1 + (-68 - 117. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+214T+5.06e4T2 1 + 214T + 5.06e4T^{2}
41 1+(75+129.i)T+(−3.44e4+5.96e4i)T2 1 + (75 + 129. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(−146+252.i)T+(−3.97e4−6.88e4i)T2 1 + (-146 + 252. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+(36−62.3i)T+(−5.19e4−8.99e4i)T2 1 + (36 - 62.3i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1−414T+1.48e5T2 1 - 414T + 1.48e5T^{2}
59 1+(372+644.i)T+(−1.02e5+1.77e5i)T2 1 + (372 + 644. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−209+361.i)T+(−1.13e5−1.96e5i)T2 1 + (-209 + 361. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(94+162.i)T+(−1.50e5+2.60e5i)T2 1 + (94 + 162. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+480T+3.57e5T2 1 + 480T + 3.57e5T^{2}
73 1−434T+3.89e5T2 1 - 434T + 3.89e5T^{2}
79 1+(676−1.17e3i)T+(−2.46e5−4.26e5i)T2 1 + (676 - 1.17e3i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+(306−530.i)T+(−2.85e5−4.95e5i)T2 1 + (306 - 530. i)T + (-2.85e5 - 4.95e5i)T^{2}
89 1−30T+7.04e5T2 1 - 30T + 7.04e5T^{2}
97 1+(−143+247.i)T+(−4.56e5−7.90e5i)T2 1 + (-143 + 247. i)T + (-4.56e5 - 7.90e5i)T^{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

−8.547260882535689327380156975654, −7.991085724660757792613368335997, −6.95758157987109535433013259062, −6.55921888067703625589068338815, −5.14147078489080546128051608332, −4.28141120490515930946244434280, −4.01645323185992284302894424068, −2.31247405426533283606355487836, −1.37140440519123501488102033210, −0.20544300134166669683436031814, 1.30650910786750581001073580124, 2.54985571187617264689192164446, 3.21065100287582212912150922592, 4.48188421458624591445003424693, 5.44787944422769105583285265215, 5.96356101105182161632133712485, 6.98870363129822896151130558666, 8.025601777667353844490279152195, 8.540068900937614714903440636934, 9.110082105687472034628756895982

Graph of the ZZ-function along the critical line