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Properties

Label 2-600-1.1-c3-0-18
Degree 22
Conductor 600600
Sign −1-1
Analytic cond. 35.401135.4011
Root an. cond. 5.949885.94988
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  − 3·3-s − 8·7-s + 9·9-s + 20·11-s − 22·13-s + 14·17-s + 76·19-s + 24·21-s − 56·23-s − 27·27-s − 154·29-s + 160·31-s − 60·33-s + 162·37-s + 66·39-s − 390·41-s − 388·43-s + 544·47-s − 279·49-s − 42·51-s + 210·53-s − 228·57-s − 380·59-s − 794·61-s − 72·63-s + 148·67-s + 168·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.431·7-s + 1/3·9-s + 0.548·11-s − 0.469·13-s + 0.199·17-s + 0.917·19-s + 0.249·21-s − 0.507·23-s − 0.192·27-s − 0.986·29-s + 0.926·31-s − 0.316·33-s + 0.719·37-s + 0.270·39-s − 1.48·41-s − 1.37·43-s + 1.68·47-s − 0.813·49-s − 0.115·51-s + 0.544·53-s − 0.529·57-s − 0.838·59-s − 1.66·61-s − 0.143·63-s + 0.269·67-s + 0.293·69-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 35.401135.4011
Root analytic conductor: 5.949885.94988
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 600, ( :3/2), −1)(2,\ 600,\ (\ :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
3 1+pT 1 + p T
5 1 1
good7 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
11 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
13 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
17 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
19 1−4pT+p3T2 1 - 4 p T + p^{3} T^{2}
23 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
29 1+154T+p3T2 1 + 154 T + p^{3} T^{2}
31 1−160T+p3T2 1 - 160 T + p^{3} T^{2}
37 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
41 1+390T+p3T2 1 + 390 T + p^{3} T^{2}
43 1+388T+p3T2 1 + 388 T + p^{3} T^{2}
47 1−544T+p3T2 1 - 544 T + p^{3} T^{2}
53 1−210T+p3T2 1 - 210 T + p^{3} T^{2}
59 1+380T+p3T2 1 + 380 T + p^{3} T^{2}
61 1+794T+p3T2 1 + 794 T + p^{3} T^{2}
67 1−148T+p3T2 1 - 148 T + p^{3} T^{2}
71 1+840T+p3T2 1 + 840 T + p^{3} T^{2}
73 1+858T+p3T2 1 + 858 T + p^{3} T^{2}
79 1−144T+p3T2 1 - 144 T + p^{3} T^{2}
83 1+316T+p3T2 1 + 316 T + p^{3} T^{2}
89 1−1098T+p3T2 1 - 1098 T + p^{3} T^{2}
97 1+994T+p3T2 1 + 994 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

−9.858195039064007808083291812846, −9.152756785828852556381630175871, −7.937480225642516303265012801209, −7.04398509773126147377112777179, −6.17434168605816562953460698531, −5.26760482819788343188426144623, −4.18261496955300194405272257579, −3.02016264350841949893002920227, −1.45650826873907689563251131852, 0, 1.45650826873907689563251131852, 3.02016264350841949893002920227, 4.18261496955300194405272257579, 5.26760482819788343188426144623, 6.17434168605816562953460698531, 7.04398509773126147377112777179, 7.937480225642516303265012801209, 9.152756785828852556381630175871, 9.858195039064007808083291812846

Graph of the ZZ-function along the critical line