Properties

Label 2-1-1.1-c43-0-2
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $11.7110$
Root an. cond. $3.42213$
Motivic weight $43$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.62e6·2-s + 1.01e10·3-s + 4.35e12·4-s + 6.19e14·5-s + 3.68e16·6-s + 2.58e18·7-s − 1.60e19·8-s − 2.25e20·9-s + 2.24e21·10-s + 2.74e22·11-s + 4.42e22·12-s + 6.62e23·13-s + 9.38e24·14-s + 6.29e24·15-s − 9.67e25·16-s + 5.82e25·17-s − 8.16e26·18-s − 1.98e27·19-s + 2.70e27·20-s + 2.62e28·21-s + 9.94e28·22-s − 2.61e29·23-s − 1.63e29·24-s − 7.52e29·25-s + 2.40e30·26-s − 5.61e30·27-s + 1.12e31·28-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.560·3-s + 0.495·4-s + 0.581·5-s + 0.685·6-s + 1.75·7-s − 0.616·8-s − 0.686·9-s + 0.710·10-s + 1.11·11-s + 0.277·12-s + 0.743·13-s + 2.14·14-s + 0.325·15-s − 1.24·16-s + 0.204·17-s − 0.839·18-s − 0.636·19-s + 0.288·20-s + 0.980·21-s + 1.36·22-s − 1.37·23-s − 0.345·24-s − 0.662·25-s + 0.909·26-s − 0.944·27-s + 0.867·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(44-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+43/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(11.7110\)
Root analytic conductor: \(3.42213\)
Motivic weight: \(43\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :43/2),\ 1)\)

Particular Values

\(L(22)\) \(\approx\) \(4.325994499\)
\(L(\frac12)\) \(\approx\) \(4.325994499\)
\(L(\frac{45}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 3.62e6T + 8.79e12T^{2} \)
3 \( 1 - 1.01e10T + 3.28e20T^{2} \)
5 \( 1 - 6.19e14T + 1.13e30T^{2} \)
7 \( 1 - 2.58e18T + 2.18e36T^{2} \)
11 \( 1 - 2.74e22T + 6.02e44T^{2} \)
13 \( 1 - 6.62e23T + 7.93e47T^{2} \)
17 \( 1 - 5.82e25T + 8.11e52T^{2} \)
19 \( 1 + 1.98e27T + 9.69e54T^{2} \)
23 \( 1 + 2.61e29T + 3.58e58T^{2} \)
29 \( 1 - 2.27e31T + 7.64e62T^{2} \)
31 \( 1 + 1.73e32T + 1.34e64T^{2} \)
37 \( 1 - 5.95e32T + 2.70e67T^{2} \)
41 \( 1 + 5.56e34T + 2.23e69T^{2} \)
43 \( 1 - 1.13e35T + 1.73e70T^{2} \)
47 \( 1 - 2.48e35T + 7.94e71T^{2} \)
53 \( 1 - 3.93e36T + 1.39e74T^{2} \)
59 \( 1 - 6.06e37T + 1.40e76T^{2} \)
61 \( 1 - 3.31e38T + 5.87e76T^{2} \)
67 \( 1 - 2.46e38T + 3.32e78T^{2} \)
71 \( 1 + 5.22e39T + 4.01e79T^{2} \)
73 \( 1 + 1.32e40T + 1.32e80T^{2} \)
79 \( 1 + 1.99e40T + 3.96e81T^{2} \)
83 \( 1 + 5.56e39T + 3.31e82T^{2} \)
89 \( 1 + 2.64e41T + 6.66e83T^{2} \)
97 \( 1 - 3.46e42T + 2.69e85T^{2} \)
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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

−21.78623200245439295075697197117, −20.51071314133364406788961231168, −17.70121612920923324856198162370, −14.64298428300446705778482044168, −13.90770841434162502850369730943, −11.62630428838795880795745896470, −8.611581403957278931214567809049, −5.74213419822693390400659725987, −4.00583767473095125197890886134, −1.93411219675686423614527871698, 1.93411219675686423614527871698, 4.00583767473095125197890886134, 5.74213419822693390400659725987, 8.611581403957278931214567809049, 11.62630428838795880795745896470, 13.90770841434162502850369730943, 14.64298428300446705778482044168, 17.70121612920923324856198162370, 20.51071314133364406788961231168, 21.78623200245439295075697197117

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