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Properties

Label 2-1620-9.4-c1-0-5
Degree 22
Conductor 16201620
Sign 0.766−0.642i0.766 - 0.642i
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 + (2 + 3.46i)7-s + (−1.5 − 2.59i)11-s + (2 − 3.46i)13-s + 5·19-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s + (4.5 + 7.79i)29-s + (−2.5 + 4.33i)31-s − 3.99·35-s + 2·37-s + (4.5 − 7.79i)41-s + (5 + 8.66i)43-s + (3 + 5.19i)47-s + (−4.49 + 7.79i)49-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.755 + 1.30i)7-s + (−0.452 − 0.783i)11-s + (0.554 − 0.960i)13-s + 1.14·19-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s + (0.835 + 1.44i)29-s + (−0.449 + 0.777i)31-s − 0.676·35-s + 0.328·37-s + (0.702 − 1.21i)41-s + (0.762 + 1.32i)43-s + (0.437 + 0.757i)47-s + (−0.642 + 1.11i)49-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: 0.766−0.642i0.766 - 0.642i
Analytic conductor: 12.935712.9357
Root analytic conductor: 3.596633.59663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1620(1081,⋅)\chi_{1620} (1081, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1620, ( :1/2), 0.766−0.642i)(2,\ 1620,\ (\ :1/2),\ 0.766 - 0.642i)

Particular Values

L(1)L(1) ≈\approx 1.8399054291.839905429
L(12)L(\frac12) ≈\approx 1.8399054291.839905429
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+(−2−3.46i)T+(−3.5+6.06i)T2 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.5+2.59i)T+(−5.5+9.52i)T2 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2+3.46i)T+(−6.5−11.2i)T2 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2}
17 1+17T2 1 + 17T^{2}
19 1−5T+19T2 1 - 5T + 19T^{2}
23 1+(−3+5.19i)T+(−11.5−19.9i)T2 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2}
29 1+(−4.5−7.79i)T+(−14.5+25.1i)T2 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2}
31 1+(2.5−4.33i)T+(−15.5−26.8i)T2 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+(−4.5+7.79i)T+(−20.5−35.5i)T2 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2}
43 1+(−5−8.66i)T+(−21.5+37.2i)T2 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2}
47 1+(−3−5.19i)T+(−23.5+40.7i)T2 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2}
53 1+12T+53T2 1 + 12T + 53T^{2}
59 1+(4.5−7.79i)T+(−29.5−51.0i)T2 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5−8.66i)T+(−30.5+52.8i)T2 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2}
67 1+(1−1.73i)T+(−33.5−58.0i)T2 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2}
71 1−3T+71T2 1 - 3T + 71T^{2}
73 1+4T+73T2 1 + 4T + 73T^{2}
79 1+(−2−3.46i)T+(−39.5+68.4i)T2 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2}
83 1+(3+5.19i)T+(−41.5+71.8i)T2 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2}
89 1+9T+89T2 1 + 9T + 89T^{2}
97 1+(1+1.73i)T+(−48.5+84.0i)T2 1 + (1 + 1.73i)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

−9.248839847856189721985601763268, −8.638352095944642759694787610973, −8.035605832404872760247455687383, −7.19097435233571141372816542410, −6.02369427964775799546206805766, −5.48974204522978763605888325413, −4.65721685080376102509392771859, −3.17044565313194721402684349304, −2.71347237714298722036415788905, −1.14293069379841707352992722486, 0.895295443662450554357463963374, 1.96205772729715479995513566826, 3.50832141578682170466398750906, 4.37140213572430849281427805456, 4.92972423493345658179987439707, 6.06673908830539985331608599105, 7.21863006699407120818014829366, 7.58586718954105188407062594631, 8.364529587754795985425252141992, 9.509584145005958275689704830079

Graph of the ZZ-function along the critical line