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Properties

Label 2-980-1.1-c1-0-12
Degree 22
Conductor 980980
Sign −1-1
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·3-s + 5-s − 2.82·9-s − 3.82·11-s − 3.58·13-s + 0.414·15-s − 6.41·17-s − 3.65·19-s + 0.585·23-s + 25-s − 2.41·27-s + 6.65·29-s − 4.58·31-s − 1.58·33-s − 3.41·37-s − 1.48·39-s − 0.585·41-s + 11.6·43-s − 2.82·45-s + 8.89·47-s − 2.65·51-s − 3.75·53-s − 3.82·55-s − 1.51·57-s − 3.41·59-s − 5.17·61-s − 3.58·65-s + ⋯
L(s)  = 1  + 0.239·3-s + 0.447·5-s − 0.942·9-s − 1.15·11-s − 0.994·13-s + 0.106·15-s − 1.55·17-s − 0.838·19-s + 0.122·23-s + 0.200·25-s − 0.464·27-s + 1.23·29-s − 0.823·31-s − 0.276·33-s − 0.561·37-s − 0.237·39-s − 0.0914·41-s + 1.77·43-s − 0.421·45-s + 1.29·47-s − 0.372·51-s − 0.516·53-s − 0.516·55-s − 0.200·57-s − 0.444·59-s − 0.662·61-s − 0.444·65-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 980, ( :1/2), −1)(2,\ 980,\ (\ :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 1
5 1−T 1 - T
7 1 1
good3 1−0.414T+3T2 1 - 0.414T + 3T^{2}
11 1+3.82T+11T2 1 + 3.82T + 11T^{2}
13 1+3.58T+13T2 1 + 3.58T + 13T^{2}
17 1+6.41T+17T2 1 + 6.41T + 17T^{2}
19 1+3.65T+19T2 1 + 3.65T + 19T^{2}
23 1−0.585T+23T2 1 - 0.585T + 23T^{2}
29 1−6.65T+29T2 1 - 6.65T + 29T^{2}
31 1+4.58T+31T2 1 + 4.58T + 31T^{2}
37 1+3.41T+37T2 1 + 3.41T + 37T^{2}
41 1+0.585T+41T2 1 + 0.585T + 41T^{2}
43 1−11.6T+43T2 1 - 11.6T + 43T^{2}
47 1−8.89T+47T2 1 - 8.89T + 47T^{2}
53 1+3.75T+53T2 1 + 3.75T + 53T^{2}
59 1+3.41T+59T2 1 + 3.41T + 59T^{2}
61 1+5.17T+61T2 1 + 5.17T + 61T^{2}
67 1+11.0T+67T2 1 + 11.0T + 67T^{2}
71 1−6.48T+71T2 1 - 6.48T + 71T^{2}
73 1+5.17T+73T2 1 + 5.17T + 73T^{2}
79 1−13.1T+79T2 1 - 13.1T + 79T^{2}
83 1−8T+83T2 1 - 8T + 83T^{2}
89 1+16.9T+89T2 1 + 16.9T + 89T^{2}
97 1+15.7T+97T2 1 + 15.7T + 97T^{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

−9.436586401103624970366121579536, −8.817887005457252545901925322918, −8.001682359547254971705465851892, −7.07028403257897782882513009419, −6.10348288756034816076782637821, −5.22370344588509756500589218610, −4.34560793882724563776534950263, −2.79899269739276673639862887077, −2.24009351742173665139724222091, 0, 2.24009351742173665139724222091, 2.79899269739276673639862887077, 4.34560793882724563776534950263, 5.22370344588509756500589218610, 6.10348288756034816076782637821, 7.07028403257897782882513009419, 8.001682359547254971705465851892, 8.817887005457252545901925322918, 9.436586401103624970366121579536

Graph of the ZZ-function along the critical line