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Properties

Label 2-1344-1.1-c3-0-45
Degree 22
Conductor 13441344
Sign −1-1
Analytic cond. 79.298579.2985
Root an. cond. 8.904978.90497
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 6·5-s − 7·7-s + 9·9-s + 36·11-s − 62·13-s + 18·15-s + 114·17-s − 76·19-s + 21·21-s + 24·23-s − 89·25-s − 27·27-s − 54·29-s + 112·31-s − 108·33-s + 42·35-s + 178·37-s + 186·39-s + 378·41-s − 172·43-s − 54·45-s + 192·47-s + 49·49-s − 342·51-s + 402·53-s − 216·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.536·5-s − 0.377·7-s + 1/3·9-s + 0.986·11-s − 1.32·13-s + 0.309·15-s + 1.62·17-s − 0.917·19-s + 0.218·21-s + 0.217·23-s − 0.711·25-s − 0.192·27-s − 0.345·29-s + 0.648·31-s − 0.569·33-s + 0.202·35-s + 0.790·37-s + 0.763·39-s + 1.43·41-s − 0.609·43-s − 0.178·45-s + 0.595·47-s + 1/7·49-s − 0.939·51-s + 1.04·53-s − 0.529·55-s + ⋯

Functional equation

Λ(s)=(1344s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1344s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13441344    =    26⋅3⋅72^{6} \cdot 3 \cdot 7
Sign: −1-1
Analytic conductor: 79.298579.2985
Root analytic conductor: 8.904978.90497
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1344, ( :3/2), −1)(2,\ 1344,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT 1 + p T
7 1+pT 1 + p T
good5 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
11 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
13 1+62T+p3T2 1 + 62 T + p^{3} T^{2}
17 1−114T+p3T2 1 - 114 T + p^{3} T^{2}
19 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
23 1−24T+p3T2 1 - 24 T + p^{3} T^{2}
29 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
31 1−112T+p3T2 1 - 112 T + p^{3} T^{2}
37 1−178T+p3T2 1 - 178 T + p^{3} T^{2}
41 1−378T+p3T2 1 - 378 T + p^{3} T^{2}
43 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
47 1−192T+p3T2 1 - 192 T + p^{3} T^{2}
53 1−402T+p3T2 1 - 402 T + p^{3} T^{2}
59 1−396T+p3T2 1 - 396 T + p^{3} T^{2}
61 1+254T+p3T2 1 + 254 T + p^{3} T^{2}
67 1+1012T+p3T2 1 + 1012 T + p^{3} T^{2}
71 1+840T+p3T2 1 + 840 T + p^{3} T^{2}
73 1−890T+p3T2 1 - 890 T + p^{3} T^{2}
79 1+80T+p3T2 1 + 80 T + p^{3} T^{2}
83 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
89 1+1638T+p3T2 1 + 1638 T + p^{3} T^{2}
97 1−1010T+p3T2 1 - 1010 T + p^{3} T^{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

−8.939755792339478494688591274786, −7.80427577876441212149525384589, −7.27308987493018330276006130476, −6.31340287693856504644574023534, −5.56961667517351518942981504657, −4.49151689438157141560488936034, −3.77300219946219926365768494190, −2.57341063364908646229312152080, −1.14363493300985782786338519497, 0, 1.14363493300985782786338519497, 2.57341063364908646229312152080, 3.77300219946219926365768494190, 4.49151689438157141560488936034, 5.56961667517351518942981504657, 6.31340287693856504644574023534, 7.27308987493018330276006130476, 7.80427577876441212149525384589, 8.939755792339478494688591274786

Graph of the ZZ-function along the critical line