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Properties

Label 2-3240-1.1-c1-0-14
Degree 22
Conductor 32403240
Sign 11
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 0.867·7-s − 4.08·11-s + 1.21·13-s + 7.82·17-s − 4.51·19-s + 2.86·23-s + 25-s + 6.29·29-s − 2.52·31-s + 0.867·35-s − 0.523·37-s − 8.12·41-s + 3.82·43-s − 1.39·47-s − 6.24·49-s + 13.9·53-s − 4.08·55-s + 6.87·59-s + 9.08·61-s + 1.21·65-s + 3.37·67-s + 3.21·71-s + 8.60·73-s − 3.54·77-s + 16.2·79-s + 6.44·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.327·7-s − 1.23·11-s + 0.336·13-s + 1.89·17-s − 1.03·19-s + 0.597·23-s + 0.200·25-s + 1.16·29-s − 0.453·31-s + 0.146·35-s − 0.0859·37-s − 1.26·41-s + 0.582·43-s − 0.202·47-s − 0.892·49-s + 1.91·53-s − 0.551·55-s + 0.894·59-s + 1.16·61-s + 0.150·65-s + 0.412·67-s + 0.381·71-s + 1.00·73-s − 0.404·77-s + 1.82·79-s + 0.707·83-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 11
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), 1)(2,\ 3240,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0777846792.077784679
L(12)L(\frac12) ≈\approx 2.0777846792.077784679
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−T 1 - T
good7 1−0.867T+7T2 1 - 0.867T + 7T^{2}
11 1+4.08T+11T2 1 + 4.08T + 11T^{2}
13 1−1.21T+13T2 1 - 1.21T + 13T^{2}
17 1−7.82T+17T2 1 - 7.82T + 17T^{2}
19 1+4.51T+19T2 1 + 4.51T + 19T^{2}
23 1−2.86T+23T2 1 - 2.86T + 23T^{2}
29 1−6.29T+29T2 1 - 6.29T + 29T^{2}
31 1+2.52T+31T2 1 + 2.52T + 31T^{2}
37 1+0.523T+37T2 1 + 0.523T + 37T^{2}
41 1+8.12T+41T2 1 + 8.12T + 41T^{2}
43 1−3.82T+43T2 1 - 3.82T + 43T^{2}
47 1+1.39T+47T2 1 + 1.39T + 47T^{2}
53 1−13.9T+53T2 1 - 13.9T + 53T^{2}
59 1−6.87T+59T2 1 - 6.87T + 59T^{2}
61 1−9.08T+61T2 1 - 9.08T + 61T^{2}
67 1−3.37T+67T2 1 - 3.37T + 67T^{2}
71 1−3.21T+71T2 1 - 3.21T + 71T^{2}
73 1−8.60T+73T2 1 - 8.60T + 73T^{2}
79 1−16.2T+79T2 1 - 16.2T + 79T^{2}
83 1−6.44T+83T2 1 - 6.44T + 83T^{2}
89 1+10.2T+89T2 1 + 10.2T + 89T^{2}
97 1+8.77T+97T2 1 + 8.77T + 97T^{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

−8.385131721213605647722104278738, −8.136714968489790950006321366925, −7.16760604124014672362937337858, −6.40043546381305178489131900913, −5.37694125650044841133974329615, −5.14170540836934730154882498559, −3.90508909815520774141186752130, −2.98207464108058208539122620428, −2.08731817679573155969198663590, −0.882826897293640083917944520406, 0.882826897293640083917944520406, 2.08731817679573155969198663590, 2.98207464108058208539122620428, 3.90508909815520774141186752130, 5.14170540836934730154882498559, 5.37694125650044841133974329615, 6.40043546381305178489131900913, 7.16760604124014672362937337858, 8.136714968489790950006321366925, 8.385131721213605647722104278738

Graph of the ZZ-function along the critical line