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Properties

Label 2-7650-1.1-c1-0-21
Degree 22
Conductor 76507650
Sign 11
Analytic cond. 61.085561.0855
Root an. cond. 7.815727.81572
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 − 1.07·7-s − 8-s + 4·11-s − 5.41·13-s + 1.07·14-s + 16-s − 17-s + 8.68·19-s − 4·22-s − 0.921·23-s + 5.41·26-s − 1.07·28-s − 4.34·29-s + 3.07·31-s − 32-s + 34-s + 8.34·37-s − 8.68·38-s + 3.26·41-s − 9.41·43-s + 4·44-s + 0.921·46-s − 2.15·47-s − 5.83·49-s − 5.41·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.407·7-s − 0.353·8-s + 1.20·11-s − 1.50·13-s + 0.288·14-s + 0.250·16-s − 0.242·17-s + 1.99·19-s − 0.852·22-s − 0.192·23-s + 1.06·26-s − 0.203·28-s − 0.805·29-s + 0.552·31-s − 0.176·32-s + 0.171·34-s + 1.37·37-s − 1.40·38-s + 0.509·41-s − 1.43·43-s + 0.603·44-s + 0.135·46-s − 0.314·47-s − 0.833·49-s − 0.751·52-s + ⋯

Functional equation

Λ(s)=(7650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76507650    =    2⋅32⋅52⋅172 \cdot 3^{2} \cdot 5^{2} \cdot 17
Sign: 11
Analytic conductor: 61.085561.0855
Root analytic conductor: 7.815727.81572
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7650, ( :1/2), 1)(2,\ 7650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2828761771.282876177
L(12)L(\frac12) ≈\approx 1.2828761771.282876177
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
17 1+T 1 + T
good7 1+1.07T+7T2 1 + 1.07T + 7T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1+5.41T+13T2 1 + 5.41T + 13T^{2}
19 1−8.68T+19T2 1 - 8.68T + 19T^{2}
23 1+0.921T+23T2 1 + 0.921T + 23T^{2}
29 1+4.34T+29T2 1 + 4.34T + 29T^{2}
31 1−3.07T+31T2 1 - 3.07T + 31T^{2}
37 1−8.34T+37T2 1 - 8.34T + 37T^{2}
41 1−3.26T+41T2 1 - 3.26T + 41T^{2}
43 1+9.41T+43T2 1 + 9.41T + 43T^{2}
47 1+2.15T+47T2 1 + 2.15T + 47T^{2}
53 1−13.5T+53T2 1 - 13.5T + 53T^{2}
59 1+10.0T+59T2 1 + 10.0T + 59T^{2}
61 1−7.23T+61T2 1 - 7.23T + 61T^{2}
67 1−2.58T+67T2 1 - 2.58T + 67T^{2}
71 1−3.60T+71T2 1 - 3.60T + 71T^{2}
73 1−6.58T+73T2 1 - 6.58T + 73T^{2}
79 1+12.4T+79T2 1 + 12.4T + 79T^{2}
83 1+8.68T+83T2 1 + 8.68T + 83T^{2}
89 1+6.15T+89T2 1 + 6.15T + 89T^{2}
97 1−2.09T+97T2 1 - 2.09T + 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

−7.79810933037071940945728585426, −7.21422422202007300433046450619, −6.72652856513421828900271145785, −5.89296293448296572319371209996, −5.15266658381571489675100972309, −4.27611416720214870378123787608, −3.35931897595406840323215442285, −2.64139414825280884533298749406, −1.63621416373466228554020977512, −0.64296210677065753136520315436, 0.64296210677065753136520315436, 1.63621416373466228554020977512, 2.64139414825280884533298749406, 3.35931897595406840323215442285, 4.27611416720214870378123787608, 5.15266658381571489675100972309, 5.89296293448296572319371209996, 6.72652856513421828900271145785, 7.21422422202007300433046450619, 7.79810933037071940945728585426

Graph of the ZZ-function along the critical line