Properties

Label 2-1-1.1-c47-0-2
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $13.9907$
Root an. cond. $3.74042$
Motivic weight $47$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.54e6·2-s + 1.52e11·3-s − 1.38e14·4-s + 4.23e16·5-s − 2.35e17·6-s − 3.90e19·7-s + 4.31e20·8-s − 3.47e21·9-s − 6.55e22·10-s + 2.66e24·11-s − 2.10e25·12-s + 2.26e26·13-s + 6.04e25·14-s + 6.44e27·15-s + 1.88e28·16-s + 8.41e28·17-s + 5.37e27·18-s + 3.16e29·19-s − 5.86e30·20-s − 5.94e30·21-s − 4.12e30·22-s + 2.26e31·23-s + 6.56e31·24-s + 1.08e33·25-s − 3.50e32·26-s − 4.57e33·27-s + 5.40e33·28-s + ⋯
L(s)  = 1  − 0.130·2-s + 0.932·3-s − 0.982·4-s + 1.59·5-s − 0.121·6-s − 0.539·7-s + 0.258·8-s − 0.130·9-s − 0.207·10-s + 0.897·11-s − 0.916·12-s + 1.50·13-s + 0.0703·14-s + 1.48·15-s + 0.949·16-s + 1.02·17-s + 0.0170·18-s + 0.281·19-s − 1.56·20-s − 0.503·21-s − 0.117·22-s + 0.226·23-s + 0.241·24-s + 1.52·25-s − 0.195·26-s − 1.05·27-s + 0.530·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(48-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+47/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(13.9907\)
Root analytic conductor: \(3.74042\)
Motivic weight: \(47\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :47/2),\ 1)\)

Particular Values

\(L(24)\) \(\approx\) \(2.466014373\)
\(L(\frac12)\) \(\approx\) \(2.466014373\)
\(L(\frac{49}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.54e6T + 1.40e14T^{2} \)
3 \( 1 - 1.52e11T + 2.65e22T^{2} \)
5 \( 1 - 4.23e16T + 7.10e32T^{2} \)
7 \( 1 + 3.90e19T + 5.24e39T^{2} \)
11 \( 1 - 2.66e24T + 8.81e48T^{2} \)
13 \( 1 - 2.26e26T + 2.26e52T^{2} \)
17 \( 1 - 8.41e28T + 6.77e57T^{2} \)
19 \( 1 - 3.16e29T + 1.26e60T^{2} \)
23 \( 1 - 2.26e31T + 1.00e64T^{2} \)
29 \( 1 + 3.39e34T + 5.40e68T^{2} \)
31 \( 1 - 1.81e35T + 1.24e70T^{2} \)
37 \( 1 + 7.13e36T + 5.07e73T^{2} \)
41 \( 1 + 2.78e37T + 6.32e75T^{2} \)
43 \( 1 + 4.92e37T + 5.92e76T^{2} \)
47 \( 1 - 1.86e39T + 3.87e78T^{2} \)
53 \( 1 + 2.48e40T + 1.09e81T^{2} \)
59 \( 1 - 3.69e40T + 1.69e83T^{2} \)
61 \( 1 - 3.52e41T + 8.13e83T^{2} \)
67 \( 1 + 1.26e42T + 6.69e85T^{2} \)
71 \( 1 + 3.26e43T + 1.02e87T^{2} \)
73 \( 1 - 3.59e43T + 3.76e87T^{2} \)
79 \( 1 + 5.82e44T + 1.54e89T^{2} \)
83 \( 1 + 3.51e44T + 1.57e90T^{2} \)
89 \( 1 + 2.92e45T + 4.18e91T^{2} \)
97 \( 1 - 2.17e46T + 2.38e93T^{2} \)
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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

−20.91124006953772812265757914752, −18.83164737581794470504404231619, −17.18891028768715635276234857165, −14.21310863853175597063482089839, −13.36363814561436126495704941728, −9.767291605655760643534174749608, −8.742156910971676986389001236815, −5.86172135153587139218720329263, −3.40086324840605483838717490068, −1.37250117030602044036451903916, 1.37250117030602044036451903916, 3.40086324840605483838717490068, 5.86172135153587139218720329263, 8.742156910971676986389001236815, 9.767291605655760643534174749608, 13.36363814561436126495704941728, 14.21310863853175597063482089839, 17.18891028768715635276234857165, 18.83164737581794470504404231619, 20.91124006953772812265757914752

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