Properties

Label 2-7e4-1.1-c3-0-137
Degree $2$
Conductor $2401$
Sign $1$
Analytic cond. $141.663$
Root an. cond. $11.9022$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.80·2-s + 4.88·3-s + 6.48·4-s − 19.4·5-s + 18.6·6-s − 5.77·8-s − 3.09·9-s − 73.8·10-s − 17.2·11-s + 31.6·12-s − 29.8·13-s − 94.9·15-s − 73.8·16-s − 33.4·17-s − 11.7·18-s + 122.·19-s − 125.·20-s − 65.8·22-s − 39.6·23-s − 28.2·24-s + 252.·25-s − 113.·26-s − 147.·27-s + 285.·29-s − 361.·30-s + 61.6·31-s − 234.·32-s + ⋯
L(s)  = 1  + 1.34·2-s + 0.940·3-s + 0.810·4-s − 1.73·5-s + 1.26·6-s − 0.255·8-s − 0.114·9-s − 2.33·10-s − 0.474·11-s + 0.762·12-s − 0.635·13-s − 1.63·15-s − 1.15·16-s − 0.476·17-s − 0.154·18-s + 1.47·19-s − 1.40·20-s − 0.637·22-s − 0.359·23-s − 0.240·24-s + 2.01·25-s − 0.855·26-s − 1.04·27-s + 1.82·29-s − 2.19·30-s + 0.357·31-s − 1.29·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2401\)    =    \(7^{4}\)
Sign: $1$
Analytic conductor: \(141.663\)
Root analytic conductor: \(11.9022\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2401,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.108065771\)
\(L(\frac12)\) \(\approx\) \(3.108065771\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 - 3.80T + 8T^{2} \)
3 \( 1 - 4.88T + 27T^{2} \)
5 \( 1 + 19.4T + 125T^{2} \)
11 \( 1 + 17.2T + 1.33e3T^{2} \)
13 \( 1 + 29.8T + 2.19e3T^{2} \)
17 \( 1 + 33.4T + 4.91e3T^{2} \)
19 \( 1 - 122.T + 6.85e3T^{2} \)
23 \( 1 + 39.6T + 1.21e4T^{2} \)
29 \( 1 - 285.T + 2.43e4T^{2} \)
31 \( 1 - 61.6T + 2.97e4T^{2} \)
37 \( 1 + 38.3T + 5.06e4T^{2} \)
41 \( 1 - 501.T + 6.89e4T^{2} \)
43 \( 1 + 83.2T + 7.95e4T^{2} \)
47 \( 1 - 26.2T + 1.03e5T^{2} \)
53 \( 1 + 428.T + 1.48e5T^{2} \)
59 \( 1 - 171.T + 2.05e5T^{2} \)
61 \( 1 - 601.T + 2.26e5T^{2} \)
67 \( 1 + 606.T + 3.00e5T^{2} \)
71 \( 1 - 309.T + 3.57e5T^{2} \)
73 \( 1 + 0.666T + 3.89e5T^{2} \)
79 \( 1 - 50.9T + 4.93e5T^{2} \)
83 \( 1 - 996.T + 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 66.6T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.436453453818466732503379174849, −7.76089198343646535881455055669, −7.21573905381182292213187244784, −6.18911894575345355500154394097, −5.08596982172528222776109369832, −4.51619058925470883818385668396, −3.72198734550940602888577897586, −3.06022022670325367765584576029, −2.50437529340574219852185894686, −0.57658418034931039680073260373, 0.57658418034931039680073260373, 2.50437529340574219852185894686, 3.06022022670325367765584576029, 3.72198734550940602888577897586, 4.51619058925470883818385668396, 5.08596982172528222776109369832, 6.18911894575345355500154394097, 7.21573905381182292213187244784, 7.76089198343646535881455055669, 8.436453453818466732503379174849

Graph of the $Z$-function along the critical line