Properties

Label 2-1-1.1-c43-0-1
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  − 4.65e6·2-s + 3.22e10·3-s + 1.28e13·4-s + 5.54e14·5-s − 1.49e17·6-s − 5.57e17·7-s − 1.89e19·8-s + 7.08e20·9-s − 2.58e21·10-s − 1.13e22·11-s + 4.14e23·12-s + 5.07e23·13-s + 2.59e24·14-s + 1.78e25·15-s − 2.51e25·16-s + 8.73e25·17-s − 3.29e27·18-s + 5.55e27·19-s + 7.12e27·20-s − 1.79e28·21-s + 5.26e28·22-s + 1.85e29·23-s − 6.08e29·24-s − 8.29e29·25-s − 2.35e30·26-s + 1.22e31·27-s − 7.16e30·28-s + ⋯
L(s)  = 1  − 1.56·2-s + 1.77·3-s + 1.46·4-s + 0.520·5-s − 2.78·6-s − 0.377·7-s − 0.724·8-s + 2.15·9-s − 0.815·10-s − 0.460·11-s + 2.59·12-s + 0.569·13-s + 0.592·14-s + 0.924·15-s − 0.324·16-s + 0.306·17-s − 3.38·18-s + 1.78·19-s + 0.760·20-s − 0.670·21-s + 0.722·22-s + 0.978·23-s − 1.28·24-s − 0.729·25-s − 0.893·26-s + 2.06·27-s − 0.551·28-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

\(L(22)\) \(\approx\) \(1.675723462\)
\(L(\frac12)\) \(\approx\) \(1.675723462\)
\(L(\frac{45}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 4.65e6T + 8.79e12T^{2} \)
3 \( 1 - 3.22e10T + 3.28e20T^{2} \)
5 \( 1 - 5.54e14T + 1.13e30T^{2} \)
7 \( 1 + 5.57e17T + 2.18e36T^{2} \)
11 \( 1 + 1.13e22T + 6.02e44T^{2} \)
13 \( 1 - 5.07e23T + 7.93e47T^{2} \)
17 \( 1 - 8.73e25T + 8.11e52T^{2} \)
19 \( 1 - 5.55e27T + 9.69e54T^{2} \)
23 \( 1 - 1.85e29T + 3.58e58T^{2} \)
29 \( 1 - 4.19e31T + 7.64e62T^{2} \)
31 \( 1 + 9.30e31T + 1.34e64T^{2} \)
37 \( 1 + 3.20e33T + 2.70e67T^{2} \)
41 \( 1 - 6.48e33T + 2.23e69T^{2} \)
43 \( 1 - 3.51e33T + 1.73e70T^{2} \)
47 \( 1 - 6.21e35T + 7.94e71T^{2} \)
53 \( 1 - 8.28e36T + 1.39e74T^{2} \)
59 \( 1 + 7.48e36T + 1.40e76T^{2} \)
61 \( 1 + 2.24e38T + 5.87e76T^{2} \)
67 \( 1 + 1.89e39T + 3.32e78T^{2} \)
71 \( 1 + 9.53e39T + 4.01e79T^{2} \)
73 \( 1 - 2.48e39T + 1.32e80T^{2} \)
79 \( 1 + 3.50e39T + 3.96e81T^{2} \)
83 \( 1 + 2.23e41T + 3.31e82T^{2} \)
89 \( 1 + 4.38e41T + 6.66e83T^{2} \)
97 \( 1 + 6.11e42T + 2.69e85T^{2} \)
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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

−20.84655842617092234587260293574, −19.56320910317706860844459680727, −18.23869234266349544116731822331, −15.85037283259849139834208795228, −13.69184573987314689104083966844, −9.992477227025999862014036048189, −8.878530349492648662914567154987, −7.46569587008960977917701953646, −2.93217208473476168019806100322, −1.36392634691248278259199015746, 1.36392634691248278259199015746, 2.93217208473476168019806100322, 7.46569587008960977917701953646, 8.878530349492648662914567154987, 9.992477227025999862014036048189, 13.69184573987314689104083966844, 15.85037283259849139834208795228, 18.23869234266349544116731822331, 19.56320910317706860844459680727, 20.84655842617092234587260293574

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