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Properties

Label 2-6069-1.1-c1-0-186
Degree 22
Conductor 60696069
Sign 11
Analytic cond. 48.461248.4612
Root an. cond. 6.961406.96140
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  + 2.35·2-s − 3-s + 3.56·4-s + 2.62·5-s − 2.35·6-s + 7-s + 3.69·8-s + 9-s + 6.18·10-s + 3.18·11-s − 3.56·12-s + 1.86·13-s + 2.35·14-s − 2.62·15-s + 1.59·16-s + 2.35·18-s + 0.108·19-s + 9.35·20-s − 21-s + 7.51·22-s + 2.37·23-s − 3.69·24-s + 1.87·25-s + 4.40·26-s − 27-s + 3.56·28-s − 4.91·29-s + ⋯
L(s)  = 1  + 1.66·2-s − 0.577·3-s + 1.78·4-s + 1.17·5-s − 0.963·6-s + 0.377·7-s + 1.30·8-s + 0.333·9-s + 1.95·10-s + 0.959·11-s − 1.02·12-s + 0.517·13-s + 0.630·14-s − 0.676·15-s + 0.398·16-s + 0.556·18-s + 0.0248·19-s + 2.09·20-s − 0.218·21-s + 1.60·22-s + 0.495·23-s − 0.755·24-s + 0.374·25-s + 0.863·26-s − 0.192·27-s + 0.674·28-s − 0.912·29-s + ⋯

Functional equation

Λ(s)=(6069s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6069s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 60696069    =    3⋅7⋅1723 \cdot 7 \cdot 17^{2}
Sign: 11
Analytic conductor: 48.461248.4612
Root analytic conductor: 6.961406.96140
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6069, ( :1/2), 1)(2,\ 6069,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.7519333656.751933365
L(12)L(\frac12) ≈\approx 6.7519333656.751933365
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)
bad3 1+T 1 + T
7 1−T 1 - T
17 1 1
good2 1−2.35T+2T2 1 - 2.35T + 2T^{2}
5 1−2.62T+5T2 1 - 2.62T + 5T^{2}
11 1−3.18T+11T2 1 - 3.18T + 11T^{2}
13 1−1.86T+13T2 1 - 1.86T + 13T^{2}
19 1−0.108T+19T2 1 - 0.108T + 19T^{2}
23 1−2.37T+23T2 1 - 2.37T + 23T^{2}
29 1+4.91T+29T2 1 + 4.91T + 29T^{2}
31 1−8.37T+31T2 1 - 8.37T + 31T^{2}
37 1−7.42T+37T2 1 - 7.42T + 37T^{2}
41 1−4.38T+41T2 1 - 4.38T + 41T^{2}
43 1+6.27T+43T2 1 + 6.27T + 43T^{2}
47 1+13.1T+47T2 1 + 13.1T + 47T^{2}
53 1−7.25T+53T2 1 - 7.25T + 53T^{2}
59 1−11.1T+59T2 1 - 11.1T + 59T^{2}
61 1+9.64T+61T2 1 + 9.64T + 61T^{2}
67 1+11.5T+67T2 1 + 11.5T + 67T^{2}
71 1−10.3T+71T2 1 - 10.3T + 71T^{2}
73 1+6.12T+73T2 1 + 6.12T + 73T^{2}
79 1−4.31T+79T2 1 - 4.31T + 79T^{2}
83 1+3.59T+83T2 1 + 3.59T + 83T^{2}
89 1+4.30T+89T2 1 + 4.30T + 89T^{2}
97 1−3.80T+97T2 1 - 3.80T + 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

−7.80350870051822767917735662491, −6.82269532565358116938425843466, −6.34258985700343447375787746828, −5.88401465368974986841797161749, −5.20774435629773845870766102074, −4.55905816095687621861038635305, −3.86727254009256434726038920404, −2.94692826006698162384884733955, −2.01360197268676995932428001037, −1.21783830891736367648374325968, 1.21783830891736367648374325968, 2.01360197268676995932428001037, 2.94692826006698162384884733955, 3.86727254009256434726038920404, 4.55905816095687621861038635305, 5.20774435629773845870766102074, 5.88401465368974986841797161749, 6.34258985700343447375787746828, 6.82269532565358116938425843466, 7.80350870051822767917735662491

Graph of the ZZ-function along the critical line