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Properties

Label 2-1620-9.7-c1-0-14
Degree 22
Conductor 16201620
Sign −0.642+0.766i-0.642 + 0.766i
Analytic cond. 12.935712.9357
Root an. cond. 3.596633.59663
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  + (−0.5 − 0.866i)5-s + (1.36 − 2.36i)7-s + (0.866 − 1.5i)11-s + (−2.73 − 4.73i)13-s + 4.73·17-s − 4.46·19-s + (1.73 + 3i)23-s + (−0.499 + 0.866i)25-s + (−3.86 + 6.69i)29-s + (−2.96 − 5.13i)31-s − 2.73·35-s − 6.19·37-s + (−5.59 − 9.69i)41-s + (−1.63 + 2.83i)43-s + (−0.633 + 1.09i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.516 − 0.894i)7-s + (0.261 − 0.452i)11-s + (−0.757 − 1.31i)13-s + 1.14·17-s − 1.02·19-s + (0.361 + 0.625i)23-s + (−0.0999 + 0.173i)25-s + (−0.717 + 1.24i)29-s + (−0.532 − 0.922i)31-s − 0.461·35-s − 1.01·37-s + (−0.874 − 1.51i)41-s + (−0.249 + 0.431i)43-s + (−0.0924 + 0.160i)47-s + ⋯

Functional equation

Λ(s)=(1620s/2 ΓC(s) L(s)=((−0.642+0.766i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1620s/2 ΓC(s+1/2) L(s)=((−0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: −0.642+0.766i-0.642 + 0.766i
Analytic conductor: 12.935712.9357
Root analytic conductor: 3.596633.59663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1620(541,⋅)\chi_{1620} (541, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1620, ( :1/2), −0.642+0.766i)(2,\ 1620,\ (\ :1/2),\ -0.642 + 0.766i)

Particular Values

L(1)L(1) ≈\approx 1.2042021041.204202104
L(12)L(\frac12) ≈\approx 1.2042021041.204202104
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
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+(−1.36+2.36i)T+(−3.5−6.06i)T2 1 + (-1.36 + 2.36i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.866+1.5i)T+(−5.5−9.52i)T2 1 + (-0.866 + 1.5i)T + (-5.5 - 9.52i)T^{2}
13 1+(2.73+4.73i)T+(−6.5+11.2i)T2 1 + (2.73 + 4.73i)T + (-6.5 + 11.2i)T^{2}
17 1−4.73T+17T2 1 - 4.73T + 17T^{2}
19 1+4.46T+19T2 1 + 4.46T + 19T^{2}
23 1+(−1.73−3i)T+(−11.5+19.9i)T2 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.86−6.69i)T+(−14.5−25.1i)T2 1 + (3.86 - 6.69i)T + (-14.5 - 25.1i)T^{2}
31 1+(2.96+5.13i)T+(−15.5+26.8i)T2 1 + (2.96 + 5.13i)T + (-15.5 + 26.8i)T^{2}
37 1+6.19T+37T2 1 + 6.19T + 37T^{2}
41 1+(5.59+9.69i)T+(−20.5+35.5i)T2 1 + (5.59 + 9.69i)T + (-20.5 + 35.5i)T^{2}
43 1+(1.63−2.83i)T+(−21.5−37.2i)T2 1 + (1.63 - 2.83i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.633−1.09i)T+(−23.5−40.7i)T2 1 + (0.633 - 1.09i)T + (-23.5 - 40.7i)T^{2}
53 1−7.26T+53T2 1 - 7.26T + 53T^{2}
59 1+(3.86+6.69i)T+(−29.5+51.0i)T2 1 + (3.86 + 6.69i)T + (-29.5 + 51.0i)T^{2}
61 1+(−2+3.46i)T+(−30.5−52.8i)T2 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.19+5.53i)T+(−33.5+58.0i)T2 1 + (3.19 + 5.53i)T + (-33.5 + 58.0i)T^{2}
71 1−11.1T+71T2 1 - 11.1T + 71T^{2}
73 1+0.196T+73T2 1 + 0.196T + 73T^{2}
79 1+(−7.19+12.4i)T+(−39.5−68.4i)T2 1 + (-7.19 + 12.4i)T + (-39.5 - 68.4i)T^{2}
83 1+(7.56−13.0i)T+(−41.5−71.8i)T2 1 + (7.56 - 13.0i)T + (-41.5 - 71.8i)T^{2}
89 1+5.19T+89T2 1 + 5.19T + 89T^{2}
97 1+(0.366−0.633i)T+(−48.5−84.0i)T2 1 + (0.366 - 0.633i)T + (-48.5 - 84.0i)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

−9.058410935713674418508015546845, −8.176383333129909123921049612641, −7.61274507021519526990620715933, −6.89550435391969659932670409428, −5.58111159542779789902313269425, −5.10862584647076982724371561991, −3.94274339970630202709366820004, −3.23429752574645418149738603844, −1.69408132961571602188841322198, −0.45586241715998568903518467537, 1.73573661814151542267623526451, 2.56060835470871805314430156297, 3.83246052561741967921598481315, 4.72045635025100401508336567684, 5.55052999413787059335797801356, 6.59873461651712085522685444979, 7.18950251848095663340745154083, 8.184709350317208955006408567364, 8.843700058052850983277104623774, 9.658481382652946378741387521913

Graph of the ZZ-function along the critical line