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Properties

Label 2-42978-1.1-c1-0-2
Degree 22
Conductor 4297842978
Sign −1-1
Analytic cond. 343.181343.181
Root an. cond. 18.525118.5251
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2·5-s − 6-s − 4·7-s + 8-s + 9-s − 2·10-s − 12-s − 13-s − 4·14-s + 2·15-s + 16-s + 6·17-s + 18-s + 19-s − 2·20-s + 4·21-s − 4·23-s − 24-s − 25-s − 26-s − 27-s − 4·28-s − 29-s + 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s + 0.872·21-s − 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s − 0.185·29-s + 0.365·30-s + ⋯

Functional equation

Λ(s)=(42978s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 42978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(42978s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 42978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4297842978    =    2⋅3⋅13⋅19⋅292 \cdot 3 \cdot 13 \cdot 19 \cdot 29
Sign: −1-1
Analytic conductor: 343.181343.181
Root analytic conductor: 18.525118.5251
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 42978, ( :1/2), −1)(2,\ 42978,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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−T 1 - T
3 1+T 1 + T
13 1+T 1 + T
19 1−T 1 - T
29 1+T 1 + T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+pT2 1 + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+14T+pT2 1 + 14 T + p 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

−15.09688910770468, −14.43753037104208, −13.81244624664274, −13.40336215615006, −12.69027702829941, −12.39867911718720, −11.91297508678669, −11.62262297527100, −10.86151874149684, −10.26092794107348, −9.772407747869918, −9.483485465365959, −8.416604163841206, −7.873093177090433, −7.403421685502794, −6.717840387571606, −6.354512112692512, −5.648283108749873, −5.281898205399340, −4.354321818126516, −3.924777015874324, −3.267127480630090, −2.905507752455362, −1.835881932614478, −0.8034173549088122, 0, 0.8034173549088122, 1.835881932614478, 2.905507752455362, 3.267127480630090, 3.924777015874324, 4.354321818126516, 5.281898205399340, 5.648283108749873, 6.354512112692512, 6.717840387571606, 7.403421685502794, 7.873093177090433, 8.416604163841206, 9.483485465365959, 9.772407747869918, 10.26092794107348, 10.86151874149684, 11.62262297527100, 11.91297508678669, 12.39867911718720, 12.69027702829941, 13.40336215615006, 13.81244624664274, 14.43753037104208, 15.09688910770468

Graph of the ZZ-function along the critical line