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Properties

Label 2-240-1.1-c7-0-24
Degree 22
Conductor 240240
Sign −1-1
Analytic cond. 74.972474.9724
Root an. cond. 8.658668.65866
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  + 27·3-s + 125·5-s − 512·7-s + 729·9-s − 5.46e3·11-s + 1.01e4·13-s + 3.37e3·15-s − 9.91e3·17-s + 1.24e4·19-s − 1.38e4·21-s − 3.36e4·23-s + 1.56e4·25-s + 1.96e4·27-s − 1.87e5·29-s + 4.25e4·31-s − 1.47e5·33-s − 6.40e4·35-s − 5.44e5·37-s + 2.74e5·39-s + 3.74e5·41-s + 5.40e5·43-s + 9.11e4·45-s − 1.33e6·47-s − 5.61e5·49-s − 2.67e5·51-s + 1.30e6·53-s − 6.82e5·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.564·7-s + 1/3·9-s − 1.23·11-s + 1.28·13-s + 0.258·15-s − 0.489·17-s + 0.415·19-s − 0.325·21-s − 0.575·23-s + 1/5·25-s + 0.192·27-s − 1.43·29-s + 0.256·31-s − 0.714·33-s − 0.252·35-s − 1.76·37-s + 0.740·39-s + 0.848·41-s + 1.03·43-s + 0.149·45-s − 1.88·47-s − 0.681·49-s − 0.282·51-s + 1.20·53-s − 0.553·55-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: −1-1
Analytic conductor: 74.972474.9724
Root analytic conductor: 8.658668.65866
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 240, ( :7/2), −1)(2,\ 240,\ (\ :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−p3T 1 - p^{3} T
5 1−p3T 1 - p^{3} T
good7 1+512T+p7T2 1 + 512 T + p^{7} T^{2}
11 1+5460T+p7T2 1 + 5460 T + p^{7} T^{2}
13 1−782pT+p7T2 1 - 782 p T + p^{7} T^{2}
17 1+9918T+p7T2 1 + 9918 T + p^{7} T^{2}
19 1−12436T+p7T2 1 - 12436 T + p^{7} T^{2}
23 1+33600T+p7T2 1 + 33600 T + p^{7} T^{2}
29 1+187914T+p7T2 1 + 187914 T + p^{7} T^{2}
31 1−42592T+p7T2 1 - 42592 T + p^{7} T^{2}
37 1+544066T+p7T2 1 + 544066 T + p^{7} T^{2}
41 1−374394T+p7T2 1 - 374394 T + p^{7} T^{2}
43 1−540532T+p7T2 1 - 540532 T + p^{7} T^{2}
47 1+1338360T+p7T2 1 + 1338360 T + p^{7} T^{2}
53 1−1308222T+p7T2 1 - 1308222 T + p^{7} T^{2}
59 1+262740T+p7T2 1 + 262740 T + p^{7} T^{2}
61 1+976330T+p7T2 1 + 976330 T + p^{7} T^{2}
67 1+3559172T+p7T2 1 + 3559172 T + p^{7} T^{2}
71 1−2673720T+p7T2 1 - 2673720 T + p^{7} T^{2}
73 1+3032134T+p7T2 1 + 3032134 T + p^{7} T^{2}
79 1−5475808T+p7T2 1 - 5475808 T + p^{7} T^{2}
83 1+2231556T+p7T2 1 + 2231556 T + p^{7} T^{2}
89 1+10050678T+p7T2 1 + 10050678 T + p^{7} T^{2}
97 1−5727554T+p7T2 1 - 5727554 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

−10.32737482471637215150734159373, −9.394815471197496491384459784730, −8.498405629393866490786675704120, −7.50797341354544861232999962217, −6.32041948911103452292749135801, −5.32315204527691294066163117124, −3.84622825285153355522476492189, −2.80585642508887432164779339835, −1.61603332443757290603817402412, 0, 1.61603332443757290603817402412, 2.80585642508887432164779339835, 3.84622825285153355522476492189, 5.32315204527691294066163117124, 6.32041948911103452292749135801, 7.50797341354544861232999962217, 8.498405629393866490786675704120, 9.394815471197496491384459784730, 10.32737482471637215150734159373

Graph of the ZZ-function along the critical line