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Properties

Label 2-450-1.1-c3-0-16
Degree 22
Conductor 450450
Sign −1-1
Analytic cond. 26.550826.5508
Root an. cond. 5.152755.15275
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  − 2·2-s + 4·4-s + 11·7-s − 8·8-s − 36·11-s + 17·13-s − 22·14-s + 16·16-s − 12·17-s − 91·19-s + 72·22-s + 60·23-s − 34·26-s + 44·28-s − 276·29-s + 191·31-s − 32·32-s + 24·34-s + 254·37-s + 182·38-s − 60·41-s − 49·43-s − 144·44-s − 120·46-s − 600·47-s − 222·49-s + 68·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.593·7-s − 0.353·8-s − 0.986·11-s + 0.362·13-s − 0.419·14-s + 1/4·16-s − 0.171·17-s − 1.09·19-s + 0.697·22-s + 0.543·23-s − 0.256·26-s + 0.296·28-s − 1.76·29-s + 1.10·31-s − 0.176·32-s + 0.121·34-s + 1.12·37-s + 0.776·38-s − 0.228·41-s − 0.173·43-s − 0.493·44-s − 0.384·46-s − 1.86·47-s − 0.647·49-s + 0.181·52-s + ⋯

Functional equation

Λ(s)=(450s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(450s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 450450    =    2⋅32⋅522 \cdot 3^{2} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 26.550826.5508
Root analytic conductor: 5.152755.15275
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 450, ( :3/2), −1)(2,\ 450,\ (\ :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+pT 1 + p T
3 1 1
5 1 1
good7 1−11T+p3T2 1 - 11 T + p^{3} T^{2}
11 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
13 1−17T+p3T2 1 - 17 T + p^{3} T^{2}
17 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
19 1+91T+p3T2 1 + 91 T + p^{3} T^{2}
23 1−60T+p3T2 1 - 60 T + p^{3} T^{2}
29 1+276T+p3T2 1 + 276 T + p^{3} T^{2}
31 1−191T+p3T2 1 - 191 T + p^{3} T^{2}
37 1−254T+p3T2 1 - 254 T + p^{3} T^{2}
41 1+60T+p3T2 1 + 60 T + p^{3} T^{2}
43 1+49T+p3T2 1 + 49 T + p^{3} T^{2}
47 1+600T+p3T2 1 + 600 T + p^{3} T^{2}
53 1−612T+p3T2 1 - 612 T + p^{3} T^{2}
59 1+744T+p3T2 1 + 744 T + p^{3} T^{2}
61 1−167T+p3T2 1 - 167 T + p^{3} T^{2}
67 1+457T+p3T2 1 + 457 T + p^{3} T^{2}
71 1+588T+p3T2 1 + 588 T + p^{3} T^{2}
73 1+970T+p3T2 1 + 970 T + p^{3} T^{2}
79 1−164T+p3T2 1 - 164 T + p^{3} T^{2}
83 1−696T+p3T2 1 - 696 T + p^{3} T^{2}
89 1+1248T+p3T2 1 + 1248 T + p^{3} T^{2}
97 1+1099T+p3T2 1 + 1099 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

−10.29305985949291089251447219666, −9.272109423615483969690508289845, −8.335209120333436994360335921914, −7.72456294151415842150104402166, −6.61294426225742464176943026584, −5.54141004990523333473422087376, −4.37730066252441880166786444336, −2.84238334534312509172488434961, −1.61796538605619749601042645936, 0, 1.61796538605619749601042645936, 2.84238334534312509172488434961, 4.37730066252441880166786444336, 5.54141004990523333473422087376, 6.61294426225742464176943026584, 7.72456294151415842150104402166, 8.335209120333436994360335921914, 9.272109423615483969690508289845, 10.29305985949291089251447219666

Graph of the ZZ-function along the critical line