Properties

Label 2-1-1.1-c39-0-2
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $9.63395$
Root an. cond. $3.10386$
Motivic weight $39$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.10e6·2-s + 2.98e9·3-s + 6.74e11·4-s + 2.54e13·5-s + 3.30e15·6-s − 5.05e16·7-s + 1.38e17·8-s + 4.87e18·9-s + 2.81e19·10-s + 7.53e19·11-s + 2.01e21·12-s + 3.07e21·13-s − 5.59e22·14-s + 7.60e22·15-s − 2.17e23·16-s − 4.56e23·17-s + 5.39e24·18-s − 1.63e24·19-s + 1.71e25·20-s − 1.51e26·21-s + 8.33e25·22-s + 2.17e26·23-s + 4.12e26·24-s − 1.17e27·25-s + 3.40e27·26-s + 2.45e27·27-s − 3.41e28·28-s + ⋯
L(s)  = 1  + 1.49·2-s + 1.48·3-s + 1.22·4-s + 0.596·5-s + 2.21·6-s − 1.67·7-s + 0.338·8-s + 1.20·9-s + 0.890·10-s + 0.371·11-s + 1.82·12-s + 0.583·13-s − 2.50·14-s + 0.885·15-s − 0.721·16-s − 0.462·17-s + 1.79·18-s − 0.189·19-s + 0.732·20-s − 2.48·21-s + 0.554·22-s + 0.606·23-s + 0.503·24-s − 0.643·25-s + 0.870·26-s + 0.301·27-s − 2.05·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(40-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+39/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(9.63395\)
Root analytic conductor: \(3.10386\)
Motivic weight: \(39\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :39/2),\ 1)\)

Particular Values

\(L(20)\) \(\approx\) \(5.191938076\)
\(L(\frac12)\) \(\approx\) \(5.191938076\)
\(L(\frac{41}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 1.10e6T + 5.49e11T^{2} \)
3 \( 1 - 2.98e9T + 4.05e18T^{2} \)
5 \( 1 - 2.54e13T + 1.81e27T^{2} \)
7 \( 1 + 5.05e16T + 9.09e32T^{2} \)
11 \( 1 - 7.53e19T + 4.11e40T^{2} \)
13 \( 1 - 3.07e21T + 2.77e43T^{2} \)
17 \( 1 + 4.56e23T + 9.71e47T^{2} \)
19 \( 1 + 1.63e24T + 7.43e49T^{2} \)
23 \( 1 - 2.17e26T + 1.28e53T^{2} \)
29 \( 1 + 6.88e27T + 1.08e57T^{2} \)
31 \( 1 - 1.62e29T + 1.45e58T^{2} \)
37 \( 1 + 1.23e30T + 1.44e61T^{2} \)
41 \( 1 - 1.60e31T + 7.91e62T^{2} \)
43 \( 1 - 1.28e32T + 5.07e63T^{2} \)
47 \( 1 + 6.09e32T + 1.62e65T^{2} \)
53 \( 1 - 1.51e33T + 1.76e67T^{2} \)
59 \( 1 + 4.73e34T + 1.15e69T^{2} \)
61 \( 1 - 7.02e34T + 4.24e69T^{2} \)
67 \( 1 + 9.34e34T + 1.64e71T^{2} \)
71 \( 1 - 2.40e36T + 1.58e72T^{2} \)
73 \( 1 + 7.72e35T + 4.67e72T^{2} \)
79 \( 1 - 1.44e37T + 1.01e74T^{2} \)
83 \( 1 - 1.12e37T + 6.98e74T^{2} \)
89 \( 1 + 1.09e38T + 1.06e76T^{2} \)
97 \( 1 - 7.84e37T + 3.04e77T^{2} \)
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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

−22.69189038615732084842531713274, −21.08496569955469773489751373870, −19.48940533787316806759697929167, −15.57314295580205807792630990978, −13.90888175216274073422076271454, −12.96396100922649545367311468419, −9.318946974243231575874117260467, −6.36775973215910596074086255632, −3.73611095458183108711647296791, −2.58530357855283309025364860921, 2.58530357855283309025364860921, 3.73611095458183108711647296791, 6.36775973215910596074086255632, 9.318946974243231575874117260467, 12.96396100922649545367311468419, 13.90888175216274073422076271454, 15.57314295580205807792630990978, 19.48940533787316806759697929167, 21.08496569955469773489751373870, 22.69189038615732084842531713274

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