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Properties

Label 2-252-1.1-c7-0-15
Degree 22
Conductor 252252
Sign −1-1
Analytic cond. 78.721078.7210
Root an. cond. 8.872488.87248
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 240·5-s + 343·7-s − 702·11-s − 3.95e3·13-s + 3.40e3·17-s − 4.90e4·19-s + 1.15e4·23-s − 2.05e4·25-s − 4.96e4·29-s − 1.13e5·31-s + 8.23e4·35-s − 6.68e4·37-s + 3.60e5·41-s − 7.65e5·43-s + 1.34e6·47-s + 1.17e5·49-s − 3.58e5·53-s − 1.68e5·55-s − 9.30e5·59-s − 1.31e6·61-s − 9.49e5·65-s + 1.89e6·67-s − 2.27e5·71-s + 7.84e5·73-s − 2.40e5·77-s − 2.10e6·79-s − 8.62e6·83-s + ⋯
L(s)  = 1  + 0.858·5-s + 0.377·7-s − 0.159·11-s − 0.499·13-s + 0.168·17-s − 1.64·19-s + 0.197·23-s − 0.262·25-s − 0.378·29-s − 0.683·31-s + 0.324·35-s − 0.217·37-s + 0.817·41-s − 1.46·43-s + 1.88·47-s + 1/7·49-s − 0.331·53-s − 0.136·55-s − 0.589·59-s − 0.743·61-s − 0.429·65-s + 0.769·67-s − 0.0755·71-s + 0.236·73-s − 0.0601·77-s − 0.479·79-s − 1.65·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 252, ( :7/2), −1)(2,\ 252,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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 1
7 1−p3T 1 - p^{3} T
good5 1−48pT+p7T2 1 - 48 p T + p^{7} T^{2}
11 1+702T+p7T2 1 + 702 T + p^{7} T^{2}
13 1+3958T+p7T2 1 + 3958 T + p^{7} T^{2}
17 1−3408T+p7T2 1 - 3408 T + p^{7} T^{2}
19 1+49036T+p7T2 1 + 49036 T + p^{7} T^{2}
23 1−11514T+p7T2 1 - 11514 T + p^{7} T^{2}
29 1+49662T+p7T2 1 + 49662 T + p^{7} T^{2}
31 1+113320T+p7T2 1 + 113320 T + p^{7} T^{2}
37 1+66886T+p7T2 1 + 66886 T + p^{7} T^{2}
41 1−360900T+p7T2 1 - 360900 T + p^{7} T^{2}
43 1+765292T+p7T2 1 + 765292 T + p^{7} T^{2}
47 1−1344876T+p7T2 1 - 1344876 T + p^{7} T^{2}
53 1+358962T+p7T2 1 + 358962 T + p^{7} T^{2}
59 1+930528T+p7T2 1 + 930528 T + p^{7} T^{2}
61 1+1318834T+p7T2 1 + 1318834 T + p^{7} T^{2}
67 1−1893464T+p7T2 1 - 1893464 T + p^{7} T^{2}
71 1+227994T+p7T2 1 + 227994 T + p^{7} T^{2}
73 1−784934T+p7T2 1 - 784934 T + p^{7} T^{2}
79 1+2100892T+p7T2 1 + 2100892 T + p^{7} T^{2}
83 1+8629308T+p7T2 1 + 8629308 T + p^{7} T^{2}
89 1+5903100T+p7T2 1 + 5903100 T + p^{7} T^{2}
97 1−773846T+p7T2 1 - 773846 T + p^{7} 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

−10.32710956583376379242033298416, −9.396063367773072340913032960223, −8.449395644226465842662660098935, −7.32387314611039918924178205954, −6.19895322353618508221340771101, −5.27237662271730489677279398532, −4.10764779194060567809777612935, −2.53696602653211729686934420842, −1.60681576002915275841966501844, 0, 1.60681576002915275841966501844, 2.53696602653211729686934420842, 4.10764779194060567809777612935, 5.27237662271730489677279398532, 6.19895322353618508221340771101, 7.32387314611039918924178205954, 8.449395644226465842662660098935, 9.396063367773072340913032960223, 10.32710956583376379242033298416

Graph of the ZZ-function along the critical line