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Properties

Label 2-2368-1.1-c1-0-38
Degree 22
Conductor 23682368
Sign −1-1
Analytic cond. 18.908518.9085
Root an. cond. 4.348394.34839
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.30·3-s − 2.30·5-s + 2.60·7-s − 1.30·9-s − 4.30·11-s + 1.30·13-s + 3·15-s + 7.21·17-s + 6·19-s − 3.39·21-s − 4.69·23-s + 0.302·25-s + 5.60·27-s − 6.69·29-s + 1.69·31-s + 5.60·33-s − 6·35-s + 37-s − 1.69·39-s − 3.30·41-s + 8.60·43-s + 3.00·45-s + 3.39·47-s − 0.211·49-s − 9.39·51-s − 7.21·53-s + 9.90·55-s + ⋯
L(s)  = 1  − 0.752·3-s − 1.02·5-s + 0.984·7-s − 0.434·9-s − 1.29·11-s + 0.361·13-s + 0.774·15-s + 1.74·17-s + 1.37·19-s − 0.740·21-s − 0.979·23-s + 0.0605·25-s + 1.07·27-s − 1.24·29-s + 0.304·31-s + 0.975·33-s − 1.01·35-s + 0.164·37-s − 0.271·39-s − 0.515·41-s + 1.31·43-s + 0.447·45-s + 0.495·47-s − 0.0301·49-s − 1.31·51-s − 0.990·53-s + 1.33·55-s + ⋯

Functional equation

Λ(s)=(2368s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2368s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23682368    =    26⋅372^{6} \cdot 37
Sign: −1-1
Analytic conductor: 18.908518.9085
Root analytic conductor: 4.348394.34839
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2368, ( :1/2), −1)(2,\ 2368,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
37 1−T 1 - T
good3 1+1.30T+3T2 1 + 1.30T + 3T^{2}
5 1+2.30T+5T2 1 + 2.30T + 5T^{2}
7 1−2.60T+7T2 1 - 2.60T + 7T^{2}
11 1+4.30T+11T2 1 + 4.30T + 11T^{2}
13 1−1.30T+13T2 1 - 1.30T + 13T^{2}
17 1−7.21T+17T2 1 - 7.21T + 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1+4.69T+23T2 1 + 4.69T + 23T^{2}
29 1+6.69T+29T2 1 + 6.69T + 29T^{2}
31 1−1.69T+31T2 1 - 1.69T + 31T^{2}
41 1+3.30T+41T2 1 + 3.30T + 41T^{2}
43 1−8.60T+43T2 1 - 8.60T + 43T^{2}
47 1−3.39T+47T2 1 - 3.39T + 47T^{2}
53 1+7.21T+53T2 1 + 7.21T + 53T^{2}
59 1−2.60T+59T2 1 - 2.60T + 59T^{2}
61 1+9.69T+61T2 1 + 9.69T + 61T^{2}
67 1+4.30T+67T2 1 + 4.30T + 67T^{2}
71 1+11.2T+71T2 1 + 11.2T + 71T^{2}
73 1+4.30T+73T2 1 + 4.30T + 73T^{2}
79 1+4.69T+79T2 1 + 4.69T + 79T^{2}
83 1−17.2T+83T2 1 - 17.2T + 83T^{2}
89 1+13.2T+89T2 1 + 13.2T + 89T^{2}
97 1−0.788T+97T2 1 - 0.788T + 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

−8.236892518577937037428452871110, −7.78490977879740658181867648544, −7.42362953664918695756571454908, −5.91390320727409415639399791165, −5.49240628919288568642473632033, −4.76425622305794922930154831191, −3.71376498573645612762292154340, −2.83320563003836383221363635042, −1.30150012601447988763520962306, 0, 1.30150012601447988763520962306, 2.83320563003836383221363635042, 3.71376498573645612762292154340, 4.76425622305794922930154831191, 5.49240628919288568642473632033, 5.91390320727409415639399791165, 7.42362953664918695756571454908, 7.78490977879740658181867648544, 8.236892518577937037428452871110

Graph of the ZZ-function along the critical line