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Properties

Label 2-14e2-1.1-c5-0-13
Degree 22
Conductor 196196
Sign −1-1
Analytic cond. 31.435231.4352
Root an. cond. 5.606715.60671
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·3-s − 54·5-s − 99·9-s + 540·11-s + 418·13-s − 648·15-s − 594·17-s − 836·19-s − 4.10e3·23-s − 209·25-s − 4.10e3·27-s − 594·29-s − 4.25e3·31-s + 6.48e3·33-s − 298·37-s + 5.01e3·39-s − 1.72e4·41-s − 1.21e4·43-s + 5.34e3·45-s + 1.29e3·47-s − 7.12e3·51-s + 1.94e4·53-s − 2.91e4·55-s − 1.00e4·57-s + 7.66e3·59-s + 3.47e4·61-s − 2.25e4·65-s + ⋯
L(s)  = 1  + 0.769·3-s − 0.965·5-s − 0.407·9-s + 1.34·11-s + 0.685·13-s − 0.743·15-s − 0.498·17-s − 0.531·19-s − 1.61·23-s − 0.0668·25-s − 1.08·27-s − 0.131·29-s − 0.795·31-s + 1.03·33-s − 0.0357·37-s + 0.528·39-s − 1.60·41-s − 0.997·43-s + 0.393·45-s + 0.0855·47-s − 0.383·51-s + 0.953·53-s − 1.29·55-s − 0.408·57-s + 0.286·59-s + 1.19·61-s − 0.662·65-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 31.435231.4352
Root analytic conductor: 5.606715.60671
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 196, ( :5/2), −1)(2,\ 196,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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
7 1 1
good3 1−4pT+p5T2 1 - 4 p T + p^{5} T^{2}
5 1+54T+p5T2 1 + 54 T + p^{5} T^{2}
11 1−540T+p5T2 1 - 540 T + p^{5} T^{2}
13 1−418T+p5T2 1 - 418 T + p^{5} T^{2}
17 1+594T+p5T2 1 + 594 T + p^{5} T^{2}
19 1+44pT+p5T2 1 + 44 p T + p^{5} T^{2}
23 1+4104T+p5T2 1 + 4104 T + p^{5} T^{2}
29 1+594T+p5T2 1 + 594 T + p^{5} T^{2}
31 1+4256T+p5T2 1 + 4256 T + p^{5} T^{2}
37 1+298T+p5T2 1 + 298 T + p^{5} T^{2}
41 1+17226T+p5T2 1 + 17226 T + p^{5} T^{2}
43 1+12100T+p5T2 1 + 12100 T + p^{5} T^{2}
47 1−1296T+p5T2 1 - 1296 T + p^{5} T^{2}
53 1−19494T+p5T2 1 - 19494 T + p^{5} T^{2}
59 1−7668T+p5T2 1 - 7668 T + p^{5} T^{2}
61 1−34738T+p5T2 1 - 34738 T + p^{5} T^{2}
67 1−21812T+p5T2 1 - 21812 T + p^{5} T^{2}
71 1+46872T+p5T2 1 + 46872 T + p^{5} T^{2}
73 1+67562T+p5T2 1 + 67562 T + p^{5} T^{2}
79 1+76912T+p5T2 1 + 76912 T + p^{5} T^{2}
83 1+67716T+p5T2 1 + 67716 T + p^{5} T^{2}
89 1+29754T+p5T2 1 + 29754 T + p^{5} T^{2}
97 1−122398T+p5T2 1 - 122398 T + p^{5} 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

−11.44685186150128413106762335950, −10.03503985661933840888228477404, −8.773279524208492506143451817734, −8.353785535572860198352928879109, −7.12887503936441446215887311078, −5.96123271535316744908512171002, −4.14992916287121959950978852690, −3.49474016109518129157168649396, −1.85204006631698210774758760246, 0, 1.85204006631698210774758760246, 3.49474016109518129157168649396, 4.14992916287121959950978852690, 5.96123271535316744908512171002, 7.12887503936441446215887311078, 8.353785535572860198352928879109, 8.773279524208492506143451817734, 10.03503985661933840888228477404, 11.44685186150128413106762335950

Graph of the ZZ-function along the critical line