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Properties

Label 2-48e2-1.1-c1-0-1
Degree 22
Conductor 23042304
Sign 11
Analytic cond. 18.397518.3975
Root an. cond. 4.289234.28923
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  − 3.46·5-s − 3.46·7-s − 6·17-s + 4·19-s − 6.92·23-s + 6.99·25-s + 3.46·29-s − 3.46·31-s + 11.9·35-s − 6.92·37-s − 6·41-s + 4·43-s + 6.92·47-s + 4.99·49-s + 3.46·53-s + 12·59-s + 6.92·61-s − 4·67-s + 6.92·71-s − 2·73-s + 10.3·79-s + 20.7·85-s + 6·89-s − 13.8·95-s − 2·97-s + 3.46·101-s + 17.3·103-s + ⋯
L(s)  = 1  − 1.54·5-s − 1.30·7-s − 1.45·17-s + 0.917·19-s − 1.44·23-s + 1.39·25-s + 0.643·29-s − 0.622·31-s + 2.02·35-s − 1.13·37-s − 0.937·41-s + 0.609·43-s + 1.01·47-s + 0.714·49-s + 0.475·53-s + 1.56·59-s + 0.887·61-s − 0.488·67-s + 0.822·71-s − 0.234·73-s + 1.16·79-s + 2.25·85-s + 0.635·89-s − 1.42·95-s − 0.203·97-s + 0.344·101-s + 1.70·103-s + ⋯

Functional equation

Λ(s)=(2304s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2304s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23042304    =    28⋅322^{8} \cdot 3^{2}
Sign: 11
Analytic conductor: 18.397518.3975
Root analytic conductor: 4.289234.28923
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2304, ( :1/2), 1)(2,\ 2304,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.59513428160.5951342816
L(12)L(\frac12) ≈\approx 0.59513428160.5951342816
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
good5 1+3.46T+5T2 1 + 3.46T + 5T^{2}
7 1+3.46T+7T2 1 + 3.46T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+6T+17T2 1 + 6T + 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+6.92T+23T2 1 + 6.92T + 23T^{2}
29 1−3.46T+29T2 1 - 3.46T + 29T^{2}
31 1+3.46T+31T2 1 + 3.46T + 31T^{2}
37 1+6.92T+37T2 1 + 6.92T + 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1−6.92T+47T2 1 - 6.92T + 47T^{2}
53 1−3.46T+53T2 1 - 3.46T + 53T^{2}
59 1−12T+59T2 1 - 12T + 59T^{2}
61 1−6.92T+61T2 1 - 6.92T + 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1−6.92T+71T2 1 - 6.92T + 71T^{2}
73 1+2T+73T2 1 + 2T + 73T^{2}
79 1−10.3T+79T2 1 - 10.3T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+2T+97T2 1 + 2T + 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

−8.872238150144058509099300033791, −8.297128464064606045304138325176, −7.33661936490695083674831742076, −6.87296624703370658242518001585, −6.00748604342774431415303044952, −4.89148071841182910687091315400, −3.88816374686518382818613341114, −3.50402801172082537256356145911, −2.34522358440884180999113886745, −0.47076910174565786246861024592, 0.47076910174565786246861024592, 2.34522358440884180999113886745, 3.50402801172082537256356145911, 3.88816374686518382818613341114, 4.89148071841182910687091315400, 6.00748604342774431415303044952, 6.87296624703370658242518001585, 7.33661936490695083674831742076, 8.297128464064606045304138325176, 8.872238150144058509099300033791

Graph of the ZZ-function along the critical line