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Properties

Label 2-450-1.1-c3-0-7
Degree 22
Conductor 450450
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s − 23·7-s + 8·8-s + 30·11-s − 29·13-s − 46·14-s + 16·16-s + 78·17-s + 149·19-s + 60·22-s + 150·23-s − 58·26-s − 92·28-s + 234·29-s − 217·31-s + 32·32-s + 156·34-s − 146·37-s + 298·38-s + 156·41-s + 433·43-s + 120·44-s + 300·46-s + 30·47-s + 186·49-s − 116·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.24·7-s + 0.353·8-s + 0.822·11-s − 0.618·13-s − 0.878·14-s + 1/4·16-s + 1.11·17-s + 1.79·19-s + 0.581·22-s + 1.35·23-s − 0.437·26-s − 0.620·28-s + 1.49·29-s − 1.25·31-s + 0.176·32-s + 0.786·34-s − 0.648·37-s + 1.27·38-s + 0.594·41-s + 1.53·43-s + 0.411·44-s + 0.961·46-s + 0.0931·47-s + 0.542·49-s − 0.309·52-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: 11
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: 00
Selberg data: (2, 450, ( :3/2), 1)(2,\ 450,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.9203166672.920316667
L(12)L(\frac12) ≈\approx 2.9203166672.920316667
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+23T+p3T2 1 + 23 T + p^{3} T^{2}
11 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
13 1+29T+p3T2 1 + 29 T + p^{3} T^{2}
17 1−78T+p3T2 1 - 78 T + p^{3} T^{2}
19 1−149T+p3T2 1 - 149 T + p^{3} T^{2}
23 1−150T+p3T2 1 - 150 T + p^{3} T^{2}
29 1−234T+p3T2 1 - 234 T + p^{3} T^{2}
31 1+7pT+p3T2 1 + 7 p T + p^{3} T^{2}
37 1+146T+p3T2 1 + 146 T + p^{3} T^{2}
41 1−156T+p3T2 1 - 156 T + p^{3} T^{2}
43 1−433T+p3T2 1 - 433 T + p^{3} T^{2}
47 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
53 1+552T+p3T2 1 + 552 T + p^{3} T^{2}
59 1−270T+p3T2 1 - 270 T + p^{3} T^{2}
61 1−275T+p3T2 1 - 275 T + p^{3} T^{2}
67 1+803T+p3T2 1 + 803 T + p^{3} T^{2}
71 1+660T+p3T2 1 + 660 T + p^{3} T^{2}
73 1−646T+p3T2 1 - 646 T + p^{3} T^{2}
79 1−992T+p3T2 1 - 992 T + p^{3} T^{2}
83 1+846T+p3T2 1 + 846 T + p^{3} T^{2}
89 1−1488T+p3T2 1 - 1488 T + p^{3} T^{2}
97 1−319T+p3T2 1 - 319 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

−10.72791457191066778887173140302, −9.715550404162922047189373314224, −9.153560049403694244995878743819, −7.57986522202087523936994107537, −6.89257208206227116290587146363, −5.89548120333738021764531113823, −4.95443726687052472657240711768, −3.57333293524272170672690575761, −2.89594104811446235744912480528, −1.03050173090979794118344008072, 1.03050173090979794118344008072, 2.89594104811446235744912480528, 3.57333293524272170672690575761, 4.95443726687052472657240711768, 5.89548120333738021764531113823, 6.89257208206227116290587146363, 7.57986522202087523936994107537, 9.153560049403694244995878743819, 9.715550404162922047189373314224, 10.72791457191066778887173140302

Graph of the ZZ-function along the critical line