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Properties

Label 2-15e2-1.1-c1-0-6
Degree 22
Conductor 225225
Sign −1-1
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5·7-s − 5·13-s + 4·16-s − 19-s + 10·28-s − 7·31-s + 10·37-s − 5·43-s + 18·49-s + 10·52-s − 13·61-s − 8·64-s − 5·67-s + 10·73-s + 2·76-s − 4·79-s + 25·91-s − 5·97-s − 20·103-s − 19·109-s − 20·112-s + ⋯
L(s)  = 1  − 4-s − 1.88·7-s − 1.38·13-s + 16-s − 0.229·19-s + 1.88·28-s − 1.25·31-s + 1.64·37-s − 0.762·43-s + 18/7·49-s + 1.38·52-s − 1.66·61-s − 64-s − 0.610·67-s + 1.17·73-s + 0.229·76-s − 0.450·79-s + 2.62·91-s − 0.507·97-s − 1.97·103-s − 1.81·109-s − 1.88·112-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 225, ( :1/2), −1)(2,\ 225,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 1 1
good2 1+pT2 1 + p T^{2}
7 1+5T+pT2 1 + 5 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+13T+pT2 1 + 13 T + p T^{2}
67 1+5T+pT2 1 + 5 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+5T+pT2 1 + 5 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

−12.11951399488920175322797663822, −10.49393585601286522315094653400, −9.599576350131101649309623215930, −9.206590206514279856778701903578, −7.75954149302713039814888795826, −6.62145632893118011347307968766, −5.46981299804941090347456591942, −4.14585724745469274814887485348, −2.93248946372060750659719660133, 0, 2.93248946372060750659719660133, 4.14585724745469274814887485348, 5.46981299804941090347456591942, 6.62145632893118011347307968766, 7.75954149302713039814888795826, 9.206590206514279856778701903578, 9.599576350131101649309623215930, 10.49393585601286522315094653400, 12.11951399488920175322797663822

Graph of the ZZ-function along the critical line