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Properties

Label 2-3240-9.4-c1-0-39
Degree 22
Conductor 32403240
Sign −0.766+0.642i-0.766 + 0.642i
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
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)11-s + (−3 + 5.19i)13-s + 6·17-s − 4·19-s + (−0.499 − 0.866i)25-s + (−1 − 1.73i)29-s + (4 − 6.92i)31-s − 2·37-s + (−3 + 5.19i)41-s + (−6 − 10.3i)43-s + (4 + 6.92i)47-s + (3.5 − 6.06i)49-s − 6·53-s − 3.99·55-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.603 − 1.04i)11-s + (−0.832 + 1.44i)13-s + 1.45·17-s − 0.917·19-s + (−0.0999 − 0.173i)25-s + (−0.185 − 0.321i)29-s + (0.718 − 1.24i)31-s − 0.328·37-s + (−0.468 + 0.811i)41-s + (−0.914 − 1.58i)43-s + (0.583 + 1.01i)47-s + (0.5 − 0.866i)49-s − 0.824·53-s − 0.539·55-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((−0.766+0.642i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=((−0.766+0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{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: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: −0.766+0.642i-0.766 + 0.642i
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3240(1081,⋅)\chi_{3240} (1081, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), −0.766+0.642i)(2,\ 3240,\ (\ :1/2),\ -0.766 + 0.642i)

Particular Values

L(1)L(1) ≈\approx 0.80429666460.8042966646
L(12)L(\frac12) ≈\approx 0.80429666460.8042966646
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+(−3.5+6.06i)T2 1 + (-3.5 + 6.06i)T^{2}
11 1+(2+3.46i)T+(−5.5+9.52i)T2 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2}
13 1+(3−5.19i)T+(−6.5−11.2i)T2 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2}
17 1−6T+17T2 1 - 6T + 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(1+1.73i)T+(−14.5+25.1i)T2 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4+6.92i)T+(−15.5−26.8i)T2 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+(3−5.19i)T+(−20.5−35.5i)T2 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2}
43 1+(6+10.3i)T+(−21.5+37.2i)T2 1 + (6 + 10.3i)T + (-21.5 + 37.2i)T^{2}
47 1+(−4−6.92i)T+(−23.5+40.7i)T2 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1+(−6+10.3i)T+(−29.5−51.0i)T2 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2}
61 1+(7+12.1i)T+(−30.5+52.8i)T2 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2}
67 1+(2−3.46i)T+(−33.5−58.0i)T2 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1+(−4−6.92i)T+(−39.5+68.4i)T2 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2}
83 1+(6+10.3i)T+(−41.5+71.8i)T2 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2}
89 1+10T+89T2 1 + 10T + 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

−8.322873086338119079293919003831, −7.76757461116773140042942922589, −6.81364482004581152325026080271, −6.06288676651208876461685178181, −5.32497155985260880778888837747, −4.53925202465065590897985155129, −3.64814705817298084049408065344, −2.60915093873376136705339227485, −1.65092101341958065858140906244, −0.23843500139855372075214458488, 1.39097051699525463643328629306, 2.62052270679392534567657702583, 3.16836673561107486553005468021, 4.40213612816041739260271274480, 5.22566176331012922434777711248, 5.77736851132505997741793556810, 6.85093745763670635652888586003, 7.48022522321709425468438581927, 8.045604733230375329004763712853, 8.910209269918978888702986569809

Graph of the ZZ-function along the critical line