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Properties

Label 2-20e2-1.1-c3-0-23
Degree 22
Conductor 400400
Sign −1-1
Analytic cond. 23.600723.6007
Root an. cond. 4.858064.85806
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  + 4·3-s − 16·7-s − 11·9-s + 60·11-s − 86·13-s − 18·17-s − 44·19-s − 64·21-s + 48·23-s − 152·27-s − 186·29-s − 176·31-s + 240·33-s − 254·37-s − 344·39-s + 186·41-s − 100·43-s + 168·47-s − 87·49-s − 72·51-s + 498·53-s − 176·57-s + 252·59-s − 58·61-s + 176·63-s − 1.03e3·67-s + 192·69-s + ⋯
L(s)  = 1  + 0.769·3-s − 0.863·7-s − 0.407·9-s + 1.64·11-s − 1.83·13-s − 0.256·17-s − 0.531·19-s − 0.665·21-s + 0.435·23-s − 1.08·27-s − 1.19·29-s − 1.01·31-s + 1.26·33-s − 1.12·37-s − 1.41·39-s + 0.708·41-s − 0.354·43-s + 0.521·47-s − 0.253·49-s − 0.197·51-s + 1.29·53-s − 0.408·57-s + 0.556·59-s − 0.121·61-s + 0.351·63-s − 1.88·67-s + 0.334·69-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 23.600723.6007
Root analytic conductor: 4.858064.85806
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 400, ( :3/2), −1)(2,\ 400,\ (\ :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)
bad2 1 1
5 1 1
good3 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
7 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
11 1−60T+p3T2 1 - 60 T + p^{3} T^{2}
13 1+86T+p3T2 1 + 86 T + p^{3} T^{2}
17 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
19 1+44T+p3T2 1 + 44 T + p^{3} T^{2}
23 1−48T+p3T2 1 - 48 T + p^{3} T^{2}
29 1+186T+p3T2 1 + 186 T + p^{3} T^{2}
31 1+176T+p3T2 1 + 176 T + p^{3} T^{2}
37 1+254T+p3T2 1 + 254 T + p^{3} T^{2}
41 1−186T+p3T2 1 - 186 T + p^{3} T^{2}
43 1+100T+p3T2 1 + 100 T + p^{3} T^{2}
47 1−168T+p3T2 1 - 168 T + p^{3} T^{2}
53 1−498T+p3T2 1 - 498 T + p^{3} T^{2}
59 1−252T+p3T2 1 - 252 T + p^{3} T^{2}
61 1+58T+p3T2 1 + 58 T + p^{3} T^{2}
67 1+1036T+p3T2 1 + 1036 T + p^{3} T^{2}
71 1+168T+p3T2 1 + 168 T + p^{3} T^{2}
73 1+506T+p3T2 1 + 506 T + p^{3} T^{2}
79 1+272T+p3T2 1 + 272 T + p^{3} T^{2}
83 1−948T+p3T2 1 - 948 T + p^{3} T^{2}
89 1+1014T+p3T2 1 + 1014 T + p^{3} T^{2}
97 1−766T+p3T2 1 - 766 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

−10.16426665220507567247807451153, −9.215984551634522830672230320001, −8.933546350909068376610617337754, −7.49369100945417328689225532375, −6.78214162283973045706533823123, −5.60785756799232395335959444254, −4.17667124406819898322039571891, −3.17944034026101807251626063054, −2.02399032199331437437601211593, 0, 2.02399032199331437437601211593, 3.17944034026101807251626063054, 4.17667124406819898322039571891, 5.60785756799232395335959444254, 6.78214162283973045706533823123, 7.49369100945417328689225532375, 8.933546350909068376610617337754, 9.215984551634522830672230320001, 10.16426665220507567247807451153

Graph of the ZZ-function along the critical line