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Properties

Label 2-1323-1.1-c1-0-12
Degree 22
Conductor 13231323
Sign 11
Analytic cond. 10.564210.5642
Root an. cond. 3.250263.25026
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 3·5-s − 6·10-s − 2·11-s − 6·13-s − 4·16-s + 3·17-s + 6·19-s + 6·20-s + 4·22-s + 8·23-s + 4·25-s + 12·26-s + 2·29-s − 6·31-s + 8·32-s − 6·34-s + 9·37-s − 12·38-s + 9·41-s − 9·43-s − 4·44-s − 16·46-s − 3·47-s − 8·50-s − 12·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.34·5-s − 1.89·10-s − 0.603·11-s − 1.66·13-s − 16-s + 0.727·17-s + 1.37·19-s + 1.34·20-s + 0.852·22-s + 1.66·23-s + 4/5·25-s + 2.35·26-s + 0.371·29-s − 1.07·31-s + 1.41·32-s − 1.02·34-s + 1.47·37-s − 1.94·38-s + 1.40·41-s − 1.37·43-s − 0.603·44-s − 2.35·46-s − 0.437·47-s − 1.13·50-s − 1.66·52-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: yes
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 0.97387240350.9738724035
L(12)L(\frac12) ≈\approx 0.97387240350.9738724035
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+pT+pT2 1 + p T + p T^{2}
5 1−3T+pT2 1 - 3 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+6T+pT2 1 + 6 T + p T^{2}
37 1−9T+pT2 1 - 9 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1+9T+pT2 1 + 9 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1−12T+pT2 1 - 12 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1−15T+pT2 1 - 15 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−6T+pT2 1 - 6 T + p 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

−9.479394193370238459103428530459, −9.289581222021464707786603330344, −7.977598033740232460345294759921, −7.44806189292664502395185090799, −6.63733359142638917146855284352, −5.40999706001583504561647394868, −4.90845254944016731114307090228, −2.96534644851181413064575814534, −2.10923233390474501409147906935, −0.913197546741854214881948839395, 0.913197546741854214881948839395, 2.10923233390474501409147906935, 2.96534644851181413064575814534, 4.90845254944016731114307090228, 5.40999706001583504561647394868, 6.63733359142638917146855284352, 7.44806189292664502395185090799, 7.977598033740232460345294759921, 9.289581222021464707786603330344, 9.479394193370238459103428530459

Graph of the ZZ-function along the critical line