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Properties

Label 2-7200-8.5-c1-0-44
Degree 22
Conductor 72007200
Sign 0.965−0.258i0.965 - 0.258i
Analytic cond. 57.492257.4922
Root an. cond. 7.582367.58236
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  + 0.732·7-s + 2i·11-s − 3.46i·13-s + 3.46·17-s + 0.535i·19-s + 6.19·23-s + 6.92i·29-s + 5.46·31-s − 2i·37-s − 1.46·41-s + 5.26i·43-s − 3.26·47-s − 6.46·49-s − 11.4i·53-s + 7.46i·59-s + ⋯
L(s)  = 1  + 0.276·7-s + 0.603i·11-s − 0.960i·13-s + 0.840·17-s + 0.122i·19-s + 1.29·23-s + 1.28i·29-s + 0.981·31-s − 0.328i·37-s − 0.228·41-s + 0.803i·43-s − 0.476·47-s − 0.923·49-s − 1.57i·53-s + 0.971i·59-s + ⋯

Functional equation

Λ(s)=(7200s/2 ΓC(s) L(s)=((0.965−0.258i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(7200s/2 ΓC(s+1/2) L(s)=((0.965−0.258i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72007200    =    25⋅32⋅522^{5} \cdot 3^{2} \cdot 5^{2}
Sign: 0.965−0.258i0.965 - 0.258i
Analytic conductor: 57.492257.4922
Root analytic conductor: 7.582367.58236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ7200(3601,⋅)\chi_{7200} (3601, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 7200, ( :1/2), 0.965−0.258i)(2,\ 7200,\ (\ :1/2),\ 0.965 - 0.258i)

Particular Values

L(1)L(1) ≈\approx 2.2009432112.200943211
L(12)L(\frac12) ≈\approx 2.2009432112.200943211
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 1
good7 1−0.732T+7T2 1 - 0.732T + 7T^{2}
11 1−2iT−11T2 1 - 2iT - 11T^{2}
13 1+3.46iT−13T2 1 + 3.46iT - 13T^{2}
17 1−3.46T+17T2 1 - 3.46T + 17T^{2}
19 1−0.535iT−19T2 1 - 0.535iT - 19T^{2}
23 1−6.19T+23T2 1 - 6.19T + 23T^{2}
29 1−6.92iT−29T2 1 - 6.92iT - 29T^{2}
31 1−5.46T+31T2 1 - 5.46T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+1.46T+41T2 1 + 1.46T + 41T^{2}
43 1−5.26iT−43T2 1 - 5.26iT - 43T^{2}
47 1+3.26T+47T2 1 + 3.26T + 47T^{2}
53 1+11.4iT−53T2 1 + 11.4iT - 53T^{2}
59 1−7.46iT−59T2 1 - 7.46iT - 59T^{2}
61 1−8.92iT−61T2 1 - 8.92iT - 61T^{2}
67 1+10.7iT−67T2 1 + 10.7iT - 67T^{2}
71 1−5.46T+71T2 1 - 5.46T + 71T^{2}
73 1+7.46T+73T2 1 + 7.46T + 73T^{2}
79 1−1.07T+79T2 1 - 1.07T + 79T^{2}
83 1−1.26iT−83T2 1 - 1.26iT - 83T^{2}
89 1+8.92T+89T2 1 + 8.92T + 89T^{2}
97 1−14.3T+97T2 1 - 14.3T + 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

−7.950191866537723788141299933678, −7.28459336672672253973631857986, −6.65737473069404178889023875957, −5.75934663024297788387983194931, −5.08407876486799013933822311377, −4.55985690545728024525193751384, −3.39685060049567578820883270883, −2.92421014296126352813731410517, −1.74409181101807960840247744029, −0.844582828836337705682473806206, 0.72127993693250806088246944841, 1.70296722530321838429599859013, 2.73830142421368923380013122667, 3.47763395559296021226023896024, 4.39513782897119132437353519885, 5.00022631833351081218365192928, 5.84829873108416047859093253909, 6.49807271510975320049162760828, 7.18328110436812171166681736976, 7.961456150166893298237478978601

Graph of the ZZ-function along the critical line