Properties

Label 2-1-1.1-c43-0-0
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  − 1.18e6·2-s − 1.79e10·3-s − 7.39e12·4-s − 6.39e14·5-s + 2.12e16·6-s − 1.72e18·7-s + 1.91e19·8-s − 6.02e18·9-s + 7.56e20·10-s + 1.05e22·11-s + 1.32e23·12-s + 1.50e24·13-s + 2.04e24·14-s + 1.14e25·15-s + 4.23e25·16-s − 5.54e26·17-s + 7.13e24·18-s − 1.99e27·19-s + 4.72e27·20-s + 3.10e28·21-s − 1.24e28·22-s + 7.48e28·23-s − 3.44e29·24-s − 7.28e29·25-s − 1.78e30·26-s + 6.00e30·27-s + 1.27e31·28-s + ⋯
L(s)  = 1  − 0.399·2-s − 0.990·3-s − 0.840·4-s − 0.599·5-s + 0.395·6-s − 1.16·7-s + 0.734·8-s − 0.0183·9-s + 0.239·10-s + 0.429·11-s + 0.832·12-s + 1.68·13-s + 0.466·14-s + 0.593·15-s + 0.547·16-s − 1.94·17-s + 0.00733·18-s − 0.640·19-s + 0.503·20-s + 1.15·21-s − 0.171·22-s + 0.395·23-s − 0.727·24-s − 0.640·25-s − 0.674·26-s + 1.00·27-s + 0.982·28-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

\(L(22)\) \(\approx\) \(0.3799864318\)
\(L(\frac12)\) \(\approx\) \(0.3799864318\)
\(L(\frac{45}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.18e6T + 8.79e12T^{2} \)
3 \( 1 + 1.79e10T + 3.28e20T^{2} \)
5 \( 1 + 6.39e14T + 1.13e30T^{2} \)
7 \( 1 + 1.72e18T + 2.18e36T^{2} \)
11 \( 1 - 1.05e22T + 6.02e44T^{2} \)
13 \( 1 - 1.50e24T + 7.93e47T^{2} \)
17 \( 1 + 5.54e26T + 8.11e52T^{2} \)
19 \( 1 + 1.99e27T + 9.69e54T^{2} \)
23 \( 1 - 7.48e28T + 3.58e58T^{2} \)
29 \( 1 + 7.64e30T + 7.64e62T^{2} \)
31 \( 1 - 1.08e31T + 1.34e64T^{2} \)
37 \( 1 - 2.93e32T + 2.70e67T^{2} \)
41 \( 1 - 7.49e34T + 2.23e69T^{2} \)
43 \( 1 - 1.23e35T + 1.73e70T^{2} \)
47 \( 1 + 5.61e35T + 7.94e71T^{2} \)
53 \( 1 - 2.77e36T + 1.39e74T^{2} \)
59 \( 1 - 1.73e38T + 1.40e76T^{2} \)
61 \( 1 + 1.16e38T + 5.87e76T^{2} \)
67 \( 1 - 9.18e38T + 3.32e78T^{2} \)
71 \( 1 + 3.76e39T + 4.01e79T^{2} \)
73 \( 1 + 9.44e39T + 1.32e80T^{2} \)
79 \( 1 - 3.86e40T + 3.96e81T^{2} \)
83 \( 1 - 1.39e41T + 3.31e82T^{2} \)
89 \( 1 - 9.05e41T + 6.66e83T^{2} \)
97 \( 1 + 1.23e42T + 2.69e85T^{2} \)
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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

−22.41256605734739650626925635219, −19.41226615559705641608712516107, −17.72496964517227546683836679015, −16.09371509460355725212768295203, −13.14168844666834405967006159329, −11.02757837482997808670131795559, −8.881283421825448913641991162095, −6.26855815566352348958119860066, −4.04483222906678438653094116852, −0.52633746964524473091865152815, 0.52633746964524473091865152815, 4.04483222906678438653094116852, 6.26855815566352348958119860066, 8.881283421825448913641991162095, 11.02757837482997808670131795559, 13.14168844666834405967006159329, 16.09371509460355725212768295203, 17.72496964517227546683836679015, 19.41226615559705641608712516107, 22.41256605734739650626925635219

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