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Properties

Label 2-3240-9.4-c1-0-34
Degree 22
Conductor 32403240
Sign 0.173+0.984i0.173 + 0.984i
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 + (0.5 + 0.866i)7-s + (1 + 1.73i)11-s + (2.5 − 4.33i)13-s − 4·17-s − 5·19-s + (1 − 1.73i)23-s + (−0.499 − 0.866i)25-s + (−5 − 8.66i)29-s + (4 − 6.92i)31-s − 0.999·35-s − 3·37-s + (−3 + 5.19i)41-s + (−2 − 3.46i)43-s + (4 + 6.92i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.188 + 0.327i)7-s + (0.301 + 0.522i)11-s + (0.693 − 1.20i)13-s − 0.970·17-s − 1.14·19-s + (0.208 − 0.361i)23-s + (−0.0999 − 0.173i)25-s + (−0.928 − 1.60i)29-s + (0.718 − 1.24i)31-s − 0.169·35-s − 0.493·37-s + (−0.468 + 0.811i)41-s + (−0.304 − 0.528i)43-s + (0.583 + 1.01i)47-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.173+0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=((0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.173+0.984i0.173 + 0.984i
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), 0.173+0.984i)(2,\ 3240,\ (\ :1/2),\ 0.173 + 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.2113923811.211392381
L(12)L(\frac12) ≈\approx 1.2113923811.211392381
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+(−0.5−0.866i)T+(−3.5+6.06i)T2 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1−1.73i)T+(−5.5+9.52i)T2 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2.5+4.33i)T+(−6.5−11.2i)T2 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2}
17 1+4T+17T2 1 + 4T + 17T^{2}
19 1+5T+19T2 1 + 5T + 19T^{2}
23 1+(−1+1.73i)T+(−11.5−19.9i)T2 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2}
29 1+(5+8.66i)T+(−14.5+25.1i)T2 1 + (5 + 8.66i)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+3T+37T2 1 + 3T + 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+(2+3.46i)T+(−21.5+37.2i)T2 1 + (2 + 3.46i)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+(−2+3.46i)T+(−29.5−51.0i)T2 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.5−4.33i)T+(−30.5+52.8i)T2 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2}
67 1+(−3.5+6.06i)T+(−33.5−58.0i)T2 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2}
71 1−6T+71T2 1 - 6T + 71T^{2}
73 1+9T+73T2 1 + 9T + 73T^{2}
79 1+(1.5+2.59i)T+(−39.5+68.4i)T2 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2}
83 1+(1+1.73i)T+(−41.5+71.8i)T2 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2}
89 1+89T2 1 + 89T^{2}
97 1+(3.5+6.06i)T+(−48.5+84.0i)T2 1 + (3.5 + 6.06i)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

−8.376409790926206178647905351340, −7.891678216095794416998142827967, −6.93410475560692207718248810314, −6.24108497250037474737755153225, −5.57640740447330907711964446747, −4.45418684630574356493718713006, −3.89230188942959940168501655324, −2.72658674335477858753098944222, −1.96939720214591320692318070281, −0.38934661454709129745777601041, 1.18961209687811799822285303567, 2.10983549793916148648753275501, 3.48797558544588749895944457065, 4.11724954176897725403269792128, 4.89094855094810274813565626061, 5.80410108684647222389025687379, 6.81546060984119447939551891464, 7.04459811558493876626876272795, 8.398943981190422521671878335549, 8.722733822840764520888617058245

Graph of the ZZ-function along the critical line