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Properties

Label 2-320892-1.1-c1-0-6
Degree 22
Conductor 320892320892
Sign 11
Analytic cond. 2562.332562.33
Root an. cond. 50.619550.6195
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  − 3-s + 3·5-s − 4·7-s + 9-s + 13-s − 3·15-s − 17-s + 4·19-s + 4·21-s + 23-s + 4·25-s − 27-s + 2·31-s − 12·35-s − 6·37-s − 39-s + 3·41-s − 8·43-s + 3·45-s − 3·47-s + 9·49-s + 51-s + 11·53-s − 4·57-s − 3·59-s + 6·61-s − 4·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.774·15-s − 0.242·17-s + 0.917·19-s + 0.872·21-s + 0.208·23-s + 4/5·25-s − 0.192·27-s + 0.359·31-s − 2.02·35-s − 0.986·37-s − 0.160·39-s + 0.468·41-s − 1.21·43-s + 0.447·45-s − 0.437·47-s + 9/7·49-s + 0.140·51-s + 1.51·53-s − 0.529·57-s − 0.390·59-s + 0.768·61-s − 0.503·63-s + ⋯

Functional equation

Λ(s)=(320892s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(320892s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320892320892    =    22⋅3⋅112⋅13⋅172^{2} \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17
Sign: 11
Analytic conductor: 2562.332562.33
Root analytic conductor: 50.619550.6195
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 320892, ( :1/2), 1)(2,\ 320892,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1742020352.174202035
L(12)L(\frac12) ≈\approx 2.1742020352.174202035
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
3 1+T 1 + T
11 1 1
13 1−T 1 - T
17 1+T 1 + T
good5 1−3T+pT2 1 - 3 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1−11T+pT2 1 - 11 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 1−13T+pT2 1 - 13 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+9T+pT2 1 + 9 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1−7T+pT2 1 - 7 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+9T+pT2 1 + 9 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

−12.76801802678267, −12.20611451896776, −11.71263223789878, −11.34565348408402, −10.59524235045683, −10.22364012677774, −10.00734780898053, −9.447429688682248, −9.198381739421252, −8.646456906939557, −8.033258056861011, −7.305644857868216, −6.799534445751589, −6.553872641557537, −6.121684657513111, −5.451636942288604, −5.404359211993955, −4.679850119720493, −3.921053397381146, −3.424864420437863, −2.940580950513905, −2.305482102877073, −1.765513453581230, −1.028362488191869, −0.4455036749540909, 0.4455036749540909, 1.028362488191869, 1.765513453581230, 2.305482102877073, 2.940580950513905, 3.424864420437863, 3.921053397381146, 4.679850119720493, 5.404359211993955, 5.451636942288604, 6.121684657513111, 6.553872641557537, 6.799534445751589, 7.305644857868216, 8.033258056861011, 8.646456906939557, 9.198381739421252, 9.447429688682248, 10.00734780898053, 10.22364012677774, 10.59524235045683, 11.34565348408402, 11.71263223789878, 12.20611451896776, 12.76801802678267

Graph of the ZZ-function along the critical line