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Properties

Label 2-2205-1.1-c3-0-135
Degree 22
Conductor 22052205
Sign −1-1
Analytic cond. 130.099130.099
Root an. cond. 11.406111.4061
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s + 5·5-s − 42·11-s − 20·13-s + 64·16-s + 66·17-s − 38·19-s − 40·20-s − 12·23-s + 25·25-s + 258·29-s − 146·31-s + 434·37-s − 282·41-s + 20·43-s + 336·44-s − 72·47-s + 160·52-s − 336·53-s − 210·55-s − 360·59-s + 682·61-s − 512·64-s − 100·65-s + 812·67-s − 528·68-s − 810·71-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 1.15·11-s − 0.426·13-s + 16-s + 0.941·17-s − 0.458·19-s − 0.447·20-s − 0.108·23-s + 1/5·25-s + 1.65·29-s − 0.845·31-s + 1.92·37-s − 1.07·41-s + 0.0709·43-s + 1.15·44-s − 0.223·47-s + 0.426·52-s − 0.870·53-s − 0.514·55-s − 0.794·59-s + 1.43·61-s − 64-s − 0.190·65-s + 1.48·67-s − 0.941·68-s − 1.35·71-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: yes
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−pT 1 - p T
7 1 1
good2 1+p3T2 1 + p^{3} T^{2}
11 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
13 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
17 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
19 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
23 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
29 1−258T+p3T2 1 - 258 T + p^{3} T^{2}
31 1+146T+p3T2 1 + 146 T + p^{3} T^{2}
37 1−434T+p3T2 1 - 434 T + p^{3} T^{2}
41 1+282T+p3T2 1 + 282 T + p^{3} T^{2}
43 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
47 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
53 1+336T+p3T2 1 + 336 T + p^{3} T^{2}
59 1+360T+p3T2 1 + 360 T + p^{3} T^{2}
61 1−682T+p3T2 1 - 682 T + p^{3} T^{2}
67 1−812T+p3T2 1 - 812 T + p^{3} T^{2}
71 1+810T+p3T2 1 + 810 T + p^{3} T^{2}
73 1−124T+p3T2 1 - 124 T + p^{3} T^{2}
79 1−1136T+p3T2 1 - 1136 T + p^{3} T^{2}
83 1−156T+p3T2 1 - 156 T + p^{3} T^{2}
89 1+1038T+p3T2 1 + 1038 T + p^{3} T^{2}
97 1+1208T+p3T2 1 + 1208 T + p^{3} 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

−8.180845144336725208950192569272, −7.86763727884578752465186768545, −6.71055478766858683568552920170, −5.74030836243527543144812071372, −5.11714673948007374960012805625, −4.42016130412106247171583368918, −3.31092781473497419716077860863, −2.41910258333400064864603127800, −1.08144668099918349951983835532, 0, 1.08144668099918349951983835532, 2.41910258333400064864603127800, 3.31092781473497419716077860863, 4.42016130412106247171583368918, 5.11714673948007374960012805625, 5.74030836243527543144812071372, 6.71055478766858683568552920170, 7.86763727884578752465186768545, 8.180845144336725208950192569272

Graph of the ZZ-function along the critical line