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Properties

Label 2-4368-1.1-c1-0-49
Degree 22
Conductor 43684368
Sign 11
Analytic cond. 34.878634.8786
Root an. cond. 5.905815.90581
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-s + 4.16·5-s + 7-s + 9-s + 4.71·11-s + 13-s + 4.16·15-s − 0.450·17-s + 0.103·19-s + 21-s − 4.42·23-s + 12.3·25-s + 27-s + 1.89·29-s − 2.55·31-s + 4.71·33-s + 4.16·35-s − 0.450·37-s + 39-s + 4.26·41-s − 4.42·43-s + 4.16·45-s − 8.61·47-s + 49-s − 0.450·51-s + 12.6·53-s + 19.6·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.86·5-s + 0.377·7-s + 0.333·9-s + 1.42·11-s + 0.277·13-s + 1.07·15-s − 0.109·17-s + 0.0236·19-s + 0.218·21-s − 0.923·23-s + 2.46·25-s + 0.192·27-s + 0.352·29-s − 0.458·31-s + 0.821·33-s + 0.703·35-s − 0.0740·37-s + 0.160·39-s + 0.666·41-s − 0.675·43-s + 0.620·45-s − 1.25·47-s + 0.142·49-s − 0.0630·51-s + 1.74·53-s + 2.64·55-s + ⋯

Functional equation

Λ(s)=(4368s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4368s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43684368    =    24⋅3⋅7⋅132^{4} \cdot 3 \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 34.878634.8786
Root analytic conductor: 5.905815.90581
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4368, ( :1/2), 1)(2,\ 4368,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.1593939514.159393951
L(12)L(\frac12) ≈\approx 4.1593939514.159393951
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−T 1 - T
7 1−T 1 - T
13 1−T 1 - T
good5 1−4.16T+5T2 1 - 4.16T + 5T^{2}
11 1−4.71T+11T2 1 - 4.71T + 11T^{2}
17 1+0.450T+17T2 1 + 0.450T + 17T^{2}
19 1−0.103T+19T2 1 - 0.103T + 19T^{2}
23 1+4.42T+23T2 1 + 4.42T + 23T^{2}
29 1−1.89T+29T2 1 - 1.89T + 29T^{2}
31 1+2.55T+31T2 1 + 2.55T + 31T^{2}
37 1+0.450T+37T2 1 + 0.450T + 37T^{2}
41 1−4.26T+41T2 1 - 4.26T + 41T^{2}
43 1+4.42T+43T2 1 + 4.42T + 43T^{2}
47 1+8.61T+47T2 1 + 8.61T + 47T^{2}
53 1−12.6T+53T2 1 - 12.6T + 53T^{2}
59 1+7.16T+59T2 1 + 7.16T + 59T^{2}
61 1+14.2T+61T2 1 + 14.2T + 61T^{2}
67 1+12.8T+67T2 1 + 12.8T + 67T^{2}
71 1−7.16T+71T2 1 - 7.16T + 71T^{2}
73 1−1.57T+73T2 1 - 1.57T + 73T^{2}
79 1−2.34T+79T2 1 - 2.34T + 79T^{2}
83 1−8.93T+83T2 1 - 8.93T + 83T^{2}
89 1−6.61T+89T2 1 - 6.61T + 89T^{2}
97 1+4.67T+97T2 1 + 4.67T + 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

−8.593790312985305814131505869791, −7.69203486300594974136712553247, −6.66843124384951480408832773143, −6.28382666553917981170316535815, −5.52171666837116827739117514921, −4.64475999839476868617412729710, −3.76225351956494245566245814105, −2.74071002283353410936321475172, −1.83344821327538226651826415780, −1.31628656141869859854913556079, 1.31628656141869859854913556079, 1.83344821327538226651826415780, 2.74071002283353410936321475172, 3.76225351956494245566245814105, 4.64475999839476868617412729710, 5.52171666837116827739117514921, 6.28382666553917981170316535815, 6.66843124384951480408832773143, 7.69203486300594974136712553247, 8.593790312985305814131505869791

Graph of the ZZ-function along the critical line