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Properties

Label 2-4050-1.1-c1-0-22
Degree 22
Conductor 40504050
Sign 11
Analytic cond. 32.339432.3394
Root an. cond. 5.686775.68677
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  − 2-s + 4-s + 4.44·7-s − 8-s + 1.44·11-s − 2.44·13-s − 4.44·14-s + 16-s + 3.89·17-s − 0.550·19-s − 1.44·22-s − 2.89·23-s + 2.44·26-s + 4.44·28-s − 6·29-s + 6.44·31-s − 32-s − 3.89·34-s + 8·37-s + 0.550·38-s − 41-s + 7.44·43-s + 1.44·44-s + 2.89·46-s + 0.449·47-s + 12.7·49-s − 2.44·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.68·7-s − 0.353·8-s + 0.437·11-s − 0.679·13-s − 1.18·14-s + 0.250·16-s + 0.945·17-s − 0.126·19-s − 0.309·22-s − 0.604·23-s + 0.480·26-s + 0.840·28-s − 1.11·29-s + 1.15·31-s − 0.176·32-s − 0.668·34-s + 1.31·37-s + 0.0893·38-s − 0.156·41-s + 1.13·43-s + 0.218·44-s + 0.427·46-s + 0.0655·47-s + 1.82·49-s − 0.339·52-s + ⋯

Functional equation

Λ(s)=(4050s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4050s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40504050    =    2⋅34⋅522 \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 32.339432.3394
Root analytic conductor: 5.686775.68677
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4050, ( :1/2), 1)(2,\ 4050,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7800931071.780093107
L(12)L(\frac12) ≈\approx 1.7800931071.780093107
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+T 1 + T
3 1 1
5 1 1
good7 1−4.44T+7T2 1 - 4.44T + 7T^{2}
11 1−1.44T+11T2 1 - 1.44T + 11T^{2}
13 1+2.44T+13T2 1 + 2.44T + 13T^{2}
17 1−3.89T+17T2 1 - 3.89T + 17T^{2}
19 1+0.550T+19T2 1 + 0.550T + 19T^{2}
23 1+2.89T+23T2 1 + 2.89T + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1−6.44T+31T2 1 - 6.44T + 31T^{2}
37 1−8T+37T2 1 - 8T + 37T^{2}
41 1+T+41T2 1 + T + 41T^{2}
43 1−7.44T+43T2 1 - 7.44T + 43T^{2}
47 1−0.449T+47T2 1 - 0.449T + 47T^{2}
53 1−8.44T+53T2 1 - 8.44T + 53T^{2}
59 1+11.2T+59T2 1 + 11.2T + 59T^{2}
61 1+0.449T+61T2 1 + 0.449T + 61T^{2}
67 1−9.44T+67T2 1 - 9.44T + 67T^{2}
71 1−2.44T+71T2 1 - 2.44T + 71T^{2}
73 1+4.79T+73T2 1 + 4.79T + 73T^{2}
79 1+7.34T+79T2 1 + 7.34T + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1−12.8T+89T2 1 - 12.8T + 89T^{2}
97 1−13T+97T2 1 - 13T + 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

−8.339563986932870424202180468562, −7.72382008668861307350916612986, −7.38215223168032834681833095580, −6.24779322808817161528656473236, −5.51590793892054510298416854158, −4.69231405209101975306355702254, −3.92371181927958158556053464890, −2.64139178383898371260436569300, −1.80484528932071689015137879459, −0.896780737417836870195025678475, 0.896780737417836870195025678475, 1.80484528932071689015137879459, 2.64139178383898371260436569300, 3.92371181927958158556053464890, 4.69231405209101975306355702254, 5.51590793892054510298416854158, 6.24779322808817161528656473236, 7.38215223168032834681833095580, 7.72382008668861307350916612986, 8.339563986932870424202180468562

Graph of the ZZ-function along the critical line