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Properties

Label 2-2205-1.1-c3-0-170
Degree 22
Conductor 22052205
Sign −1-1
Analytic cond. 130.099130.099
Root an. cond. 11.406111.4061
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·2-s − 1.43·4-s − 5·5-s − 24.1·8-s − 12.8·10-s + 6.24·11-s + 56.3·13-s − 50.4·16-s − 24.6·17-s + 90.7·19-s + 7.19·20-s + 16·22-s − 69.8·23-s + 25·25-s + 144.·26-s − 228.·29-s + 67.8·31-s + 64.2·32-s − 63.0·34-s − 58.8·37-s + 232.·38-s + 120.·40-s − 19.2·41-s + 365.·43-s − 8.98·44-s − 178.·46-s + 195.·47-s + ⋯
L(s)  = 1  + 0.905·2-s − 0.179·4-s − 0.447·5-s − 1.06·8-s − 0.405·10-s + 0.171·11-s + 1.20·13-s − 0.787·16-s − 0.350·17-s + 1.09·19-s + 0.0804·20-s + 0.155·22-s − 0.633·23-s + 0.200·25-s + 1.08·26-s − 1.46·29-s + 0.393·31-s + 0.354·32-s − 0.317·34-s − 0.261·37-s + 0.991·38-s + 0.477·40-s − 0.0731·41-s + 1.29·43-s − 0.0307·44-s − 0.573·46-s + 0.605·47-s + ⋯

Functional equation

Λ(s)=(2205s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2205s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22052205    =    32⋅5⋅723^{2} \cdot 5 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 130.099130.099
Root analytic conductor: 11.406111.4061
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2205, ( :3/2), −1)(2,\ 2205,\ (\ :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)
bad3 1 1
5 1+5T 1 + 5T
7 1 1
good2 1−2.56T+8T2 1 - 2.56T + 8T^{2}
11 1−6.24T+1.33e3T2 1 - 6.24T + 1.33e3T^{2}
13 1−56.3T+2.19e3T2 1 - 56.3T + 2.19e3T^{2}
17 1+24.6T+4.91e3T2 1 + 24.6T + 4.91e3T^{2}
19 1−90.7T+6.85e3T2 1 - 90.7T + 6.85e3T^{2}
23 1+69.8T+1.21e4T2 1 + 69.8T + 1.21e4T^{2}
29 1+228.T+2.43e4T2 1 + 228.T + 2.43e4T^{2}
31 1−67.8T+2.97e4T2 1 - 67.8T + 2.97e4T^{2}
37 1+58.8T+5.06e4T2 1 + 58.8T + 5.06e4T^{2}
41 1+19.2T+6.89e4T2 1 + 19.2T + 6.89e4T^{2}
43 1−365.T+7.95e4T2 1 - 365.T + 7.95e4T^{2}
47 1−195.T+1.03e5T2 1 - 195.T + 1.03e5T^{2}
53 1+511.T+1.48e5T2 1 + 511.T + 1.48e5T^{2}
59 1−284T+2.05e5T2 1 - 284T + 2.05e5T^{2}
61 1−123.T+2.26e5T2 1 - 123.T + 2.26e5T^{2}
67 1−144.T+3.00e5T2 1 - 144.T + 3.00e5T^{2}
71 1+73.0T+3.57e5T2 1 + 73.0T + 3.57e5T^{2}
73 1+638.T+3.89e5T2 1 + 638.T + 3.89e5T^{2}
79 1−976.T+4.93e5T2 1 - 976.T + 4.93e5T^{2}
83 1−484.T+5.71e5T2 1 - 484.T + 5.71e5T^{2}
89 1+1.01e3T+7.04e5T2 1 + 1.01e3T + 7.04e5T^{2}
97 1+1.80e3T+9.12e5T2 1 + 1.80e3T + 9.12e5T^{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

−8.313986404571865142652630448011, −7.52195641179090705481838590446, −6.52207133860112242905809928598, −5.79689757782884252790490275939, −5.11019922436989186607765222217, −4.03945007081128344518315577254, −3.69724288265100475036162569025, −2.66483754607630004393587062209, −1.23956327088675683827205019265, 0, 1.23956327088675683827205019265, 2.66483754607630004393587062209, 3.69724288265100475036162569025, 4.03945007081128344518315577254, 5.11019922436989186607765222217, 5.79689757782884252790490275939, 6.52207133860112242905809928598, 7.52195641179090705481838590446, 8.313986404571865142652630448011

Graph of the ZZ-function along the critical line