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Properties

Label 2-42e2-4.3-c0-0-1
Degree 22
Conductor 17641764
Sign 11
Analytic cond. 0.8803500.880350
Root an. cond. 0.9382700.938270
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 1.41·5-s − 8-s + 1.41·10-s − 1.41·13-s + 16-s + 1.41·17-s − 1.41·20-s + 1.00·25-s + 1.41·26-s + 2·29-s − 32-s − 1.41·34-s + 1.41·40-s + 1.41·41-s − 1.00·50-s − 1.41·52-s − 2·58-s + 1.41·61-s + 64-s + 2.00·65-s + 1.41·68-s + 1.41·73-s − 1.41·80-s − 1.41·82-s − 2.00·85-s + ⋯
L(s)  = 1  − 2-s + 4-s − 1.41·5-s − 8-s + 1.41·10-s − 1.41·13-s + 16-s + 1.41·17-s − 1.41·20-s + 1.00·25-s + 1.41·26-s + 2·29-s − 32-s − 1.41·34-s + 1.41·40-s + 1.41·41-s − 1.00·50-s − 1.41·52-s − 2·58-s + 1.41·61-s + 64-s + 2.00·65-s + 1.41·68-s + 1.41·73-s − 1.41·80-s − 1.41·82-s − 2.00·85-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 0.8803500.880350
Root analytic conductor: 0.9382700.938270
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1764(883,⋅)\chi_{1764} (883, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1764, ( :0), 1)(2,\ 1764,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.52085881550.5208588155
L(12)L(\frac12) ≈\approx 0.52085881550.5208588155
L(1)L(1) 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+T 1 + T
3 1 1
7 1 1
good5 1+1.41T+T2 1 + 1.41T + T^{2}
11 1−T2 1 - T^{2}
13 1+1.41T+T2 1 + 1.41T + T^{2}
17 1−1.41T+T2 1 - 1.41T + T^{2}
19 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
29 1−2T+T2 1 - 2T + T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1−1.41T+T2 1 - 1.41T + T^{2}
43 1−T2 1 - T^{2}
47 1−T2 1 - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−1.41T+T2 1 - 1.41T + T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1−1.41T+T2 1 - 1.41T + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+1.41T+T2 1 + 1.41T + T^{2}
97 1−1.41T+T2 1 - 1.41T + 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

−9.573851337685422654283782445352, −8.499641658719668833265027931500, −7.937278212016017953230038094861, −7.39285223842948294178162293358, −6.68520672525392673578417997868, −5.50631582440132158687708465641, −4.47532135661802338154032159259, −3.37700479978420989531265258823, −2.51730789116789760703552203771, −0.840214520965958584451282685107, 0.840214520965958584451282685107, 2.51730789116789760703552203771, 3.37700479978420989531265258823, 4.47532135661802338154032159259, 5.50631582440132158687708465641, 6.68520672525392673578417997868, 7.39285223842948294178162293358, 7.937278212016017953230038094861, 8.499641658719668833265027931500, 9.573851337685422654283782445352

Graph of the ZZ-function along the critical line