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Properties

Label 2-1323-1.1-c1-0-7
Degree 22
Conductor 13231323
Sign 11
Analytic cond. 10.564210.5642
Root an. cond. 3.250263.25026
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  − 0.874·2-s − 1.23·4-s − 0.236·5-s + 2.82·8-s + 0.206·10-s − 0.540·11-s + 0.874·13-s + 5·17-s − 4.03·19-s + 0.291·20-s + 0.472·22-s − 5.99·23-s − 4.94·25-s − 0.763·26-s + 8.61·29-s − 6.53·31-s − 5.65·32-s − 4.37·34-s + 8.70·37-s + 3.52·38-s − 0.667·40-s + 8.70·41-s − 2.23·43-s + 0.667·44-s + 5.23·46-s + 7.47·47-s + 4.32·50-s + ⋯
L(s)  = 1  − 0.618·2-s − 0.618·4-s − 0.105·5-s + 0.999·8-s + 0.0652·10-s − 0.162·11-s + 0.242·13-s + 1.21·17-s − 0.925·19-s + 0.0652·20-s + 0.100·22-s − 1.24·23-s − 0.988·25-s − 0.149·26-s + 1.59·29-s − 1.17·31-s − 0.999·32-s − 0.749·34-s + 1.43·37-s + 0.572·38-s − 0.105·40-s + 1.35·41-s − 0.340·43-s + 0.100·44-s + 0.772·46-s + 1.08·47-s + 0.611·50-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 11
Analytic conductor: 10.564210.5642
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1323, ( :1/2), 1)(2,\ 1323,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.90170542460.9017054246
L(12)L(\frac12) ≈\approx 0.90170542460.9017054246
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)
bad3 1 1
7 1 1
good2 1+0.874T+2T2 1 + 0.874T + 2T^{2}
5 1+0.236T+5T2 1 + 0.236T + 5T^{2}
11 1+0.540T+11T2 1 + 0.540T + 11T^{2}
13 1−0.874T+13T2 1 - 0.874T + 13T^{2}
17 1−5T+17T2 1 - 5T + 17T^{2}
19 1+4.03T+19T2 1 + 4.03T + 19T^{2}
23 1+5.99T+23T2 1 + 5.99T + 23T^{2}
29 1−8.61T+29T2 1 - 8.61T + 29T^{2}
31 1+6.53T+31T2 1 + 6.53T + 31T^{2}
37 1−8.70T+37T2 1 - 8.70T + 37T^{2}
41 1−8.70T+41T2 1 - 8.70T + 41T^{2}
43 1+2.23T+43T2 1 + 2.23T + 43T^{2}
47 1−7.47T+47T2 1 - 7.47T + 47T^{2}
53 1+3.16T+53T2 1 + 3.16T + 53T^{2}
59 1−13.9T+59T2 1 - 13.9T + 59T^{2}
61 1−0.540T+61T2 1 - 0.540T + 61T^{2}
67 1+6.76T+67T2 1 + 6.76T + 67T^{2}
71 1−6.73T+71T2 1 - 6.73T + 71T^{2}
73 1−13.3T+73T2 1 - 13.3T + 73T^{2}
79 1+2.52T+79T2 1 + 2.52T + 79T^{2}
83 1−4.23T+83T2 1 - 4.23T + 83T^{2}
89 1−11.7T+89T2 1 - 11.7T + 89T^{2}
97 1−5.11T+97T2 1 - 5.11T + 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

−9.667843905372531757614818080561, −8.813308102720444799374580399624, −8.009296994460561051280804566014, −7.60180939339733156772551434906, −6.30223077681206188071927542378, −5.49076877856289621015410733327, −4.39871097506363557062342686000, −3.68071334079062017918626527098, −2.18290733472934049938907800144, −0.77429173659406431581889621458, 0.77429173659406431581889621458, 2.18290733472934049938907800144, 3.68071334079062017918626527098, 4.39871097506363557062342686000, 5.49076877856289621015410733327, 6.30223077681206188071927542378, 7.60180939339733156772551434906, 8.009296994460561051280804566014, 8.813308102720444799374580399624, 9.667843905372531757614818080561

Graph of the ZZ-function along the critical line