Properties

Label 2-1-1.1-c47-0-1
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  + 4.55e6·2-s − 2.21e11·3-s − 1.20e14·4-s − 3.08e16·5-s − 1.00e18·6-s + 1.11e20·7-s − 1.18e21·8-s + 2.26e22·9-s − 1.40e23·10-s − 1.19e24·11-s + 2.66e25·12-s − 2.71e24·13-s + 5.05e26·14-s + 6.84e27·15-s + 1.14e28·16-s + 1.17e29·17-s + 1.03e29·18-s − 1.65e30·19-s + 3.70e30·20-s − 2.46e31·21-s − 5.43e30·22-s + 5.41e31·23-s + 2.63e32·24-s + 2.40e32·25-s − 1.23e31·26-s + 8.72e32·27-s − 1.33e34·28-s + ⋯
L(s)  = 1  + 0.383·2-s − 1.36·3-s − 0.852·4-s − 1.15·5-s − 0.522·6-s + 1.53·7-s − 0.710·8-s + 0.852·9-s − 0.443·10-s − 0.402·11-s + 1.16·12-s − 0.0180·13-s + 0.588·14-s + 1.57·15-s + 0.580·16-s + 1.42·17-s + 0.326·18-s − 1.47·19-s + 0.986·20-s − 2.08·21-s − 0.154·22-s + 0.541·23-s + 0.967·24-s + 0.338·25-s − 0.00691·26-s + 0.201·27-s − 1.30·28-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

\(L(24)\) \(\approx\) \(0.7875720002\)
\(L(\frac12)\) \(\approx\) \(0.7875720002\)
\(L(\frac{49}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 4.55e6T + 1.40e14T^{2} \)
3 \( 1 + 2.21e11T + 2.65e22T^{2} \)
5 \( 1 + 3.08e16T + 7.10e32T^{2} \)
7 \( 1 - 1.11e20T + 5.24e39T^{2} \)
11 \( 1 + 1.19e24T + 8.81e48T^{2} \)
13 \( 1 + 2.71e24T + 2.26e52T^{2} \)
17 \( 1 - 1.17e29T + 6.77e57T^{2} \)
19 \( 1 + 1.65e30T + 1.26e60T^{2} \)
23 \( 1 - 5.41e31T + 1.00e64T^{2} \)
29 \( 1 + 1.03e34T + 5.40e68T^{2} \)
31 \( 1 + 3.80e34T + 1.24e70T^{2} \)
37 \( 1 - 8.41e36T + 5.07e73T^{2} \)
41 \( 1 - 7.68e37T + 6.32e75T^{2} \)
43 \( 1 + 2.17e38T + 5.92e76T^{2} \)
47 \( 1 - 2.73e39T + 3.87e78T^{2} \)
53 \( 1 - 8.29e39T + 1.09e81T^{2} \)
59 \( 1 - 7.09e40T + 1.69e83T^{2} \)
61 \( 1 + 6.43e41T + 8.13e83T^{2} \)
67 \( 1 - 1.19e43T + 6.69e85T^{2} \)
71 \( 1 - 2.64e43T + 1.02e87T^{2} \)
73 \( 1 - 3.87e43T + 3.76e87T^{2} \)
79 \( 1 + 1.22e44T + 1.54e89T^{2} \)
83 \( 1 - 1.65e45T + 1.57e90T^{2} \)
89 \( 1 + 6.00e45T + 4.18e91T^{2} \)
97 \( 1 + 1.75e46T + 2.38e93T^{2} \)
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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

−21.39108425875059520251405714977, −18.51499339407793960354130164160, −17.05701108664112257105151531070, −14.81372158901683204413673199664, −12.31445719615043186233989073150, −11.03054489242947403771681137383, −8.026379096700647627059786266112, −5.40281269628152395213217208803, −4.25474528483862484526695563349, −0.68178057301413172356960766406, 0.68178057301413172356960766406, 4.25474528483862484526695563349, 5.40281269628152395213217208803, 8.026379096700647627059786266112, 11.03054489242947403771681137383, 12.31445719615043186233989073150, 14.81372158901683204413673199664, 17.05701108664112257105151531070, 18.51499339407793960354130164160, 21.39108425875059520251405714977

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