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Properties

Label 2-9280-1.1-c1-0-106
Degree 22
Conductor 92809280
Sign 11
Analytic cond. 74.101174.1011
Root an. cond. 8.608208.60820
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  + 3.11·3-s − 5-s − 0.964·7-s + 6.71·9-s − 2.75·11-s + 3.87·13-s − 3.11·15-s + 2.11·17-s + 5.82·19-s − 3.00·21-s − 2.04·23-s + 25-s + 11.5·27-s + 29-s + 2.96·31-s − 8.58·33-s + 0.964·35-s − 4.75·37-s + 12.0·39-s + 11.3·41-s + 3.27·43-s − 6.71·45-s + 0.405·47-s − 6.07·49-s + 6.58·51-s − 6.98·53-s + 2.75·55-s + ⋯
L(s)  = 1  + 1.79·3-s − 0.447·5-s − 0.364·7-s + 2.23·9-s − 0.830·11-s + 1.07·13-s − 0.804·15-s + 0.512·17-s + 1.33·19-s − 0.655·21-s − 0.425·23-s + 0.200·25-s + 2.23·27-s + 0.185·29-s + 0.532·31-s − 1.49·33-s + 0.162·35-s − 0.781·37-s + 1.93·39-s + 1.76·41-s + 0.498·43-s − 1.00·45-s + 0.0591·47-s − 0.867·49-s + 0.921·51-s − 0.960·53-s + 0.371·55-s + ⋯

Functional equation

Λ(s)=(9280s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9280s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92809280    =    26⋅5⋅292^{6} \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 74.101174.1011
Root analytic conductor: 8.608208.60820
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9280, ( :1/2), 1)(2,\ 9280,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.1544112514.154411251
L(12)L(\frac12) ≈\approx 4.1544112514.154411251
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
5 1+T 1 + T
29 1−T 1 - T
good3 1−3.11T+3T2 1 - 3.11T + 3T^{2}
7 1+0.964T+7T2 1 + 0.964T + 7T^{2}
11 1+2.75T+11T2 1 + 2.75T + 11T^{2}
13 1−3.87T+13T2 1 - 3.87T + 13T^{2}
17 1−2.11T+17T2 1 - 2.11T + 17T^{2}
19 1−5.82T+19T2 1 - 5.82T + 19T^{2}
23 1+2.04T+23T2 1 + 2.04T + 23T^{2}
31 1−2.96T+31T2 1 - 2.96T + 31T^{2}
37 1+4.75T+37T2 1 + 4.75T + 37T^{2}
41 1−11.3T+41T2 1 - 11.3T + 41T^{2}
43 1−3.27T+43T2 1 - 3.27T + 43T^{2}
47 1−0.405T+47T2 1 - 0.405T + 47T^{2}
53 1+6.98T+53T2 1 + 6.98T + 53T^{2}
59 1+0.364T+59T2 1 + 0.364T + 59T^{2}
61 1+12.1T+61T2 1 + 12.1T + 61T^{2}
67 1+1.48T+67T2 1 + 1.48T + 67T^{2}
71 1−14.5T+71T2 1 - 14.5T + 71T^{2}
73 1+2.16T+73T2 1 + 2.16T + 73T^{2}
79 1−14.1T+79T2 1 - 14.1T + 79T^{2}
83 1+11.3T+83T2 1 + 11.3T + 83T^{2}
89 1−6.08T+89T2 1 - 6.08T + 89T^{2}
97 1+2.31T+97T2 1 + 2.31T + 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

−7.914449463449915514621696128672, −7.37429492930043605401465481194, −6.52764256821701076314579669704, −5.66835177280226087181888166645, −4.71171534632721233157341114134, −3.94100656820025680503605199694, −3.23842939970030489528123864160, −2.92331481402841027094978594330, −1.90277622373425806244971794849, −0.926486399825057779221988889953, 0.926486399825057779221988889953, 1.90277622373425806244971794849, 2.92331481402841027094978594330, 3.23842939970030489528123864160, 3.94100656820025680503605199694, 4.71171534632721233157341114134, 5.66835177280226087181888166645, 6.52764256821701076314579669704, 7.37429492930043605401465481194, 7.914449463449915514621696128672

Graph of the ZZ-function along the critical line