Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 33
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.71e5·2-s + 5.34e7·3-s + 2.07e10·4-s − 2.01e11·5-s − 9.15e12·6-s + 5.50e13·7-s − 2.08e15·8-s − 2.70e15·9-s + 3.45e16·10-s − 8.18e16·11-s + 1.11e18·12-s − 1.90e18·13-s − 9.42e18·14-s − 1.07e19·15-s + 1.79e20·16-s − 3.33e20·17-s + 4.63e20·18-s − 1.40e20·19-s − 4.18e21·20-s + 2.94e21·21-s + 1.40e22·22-s + 3.12e22·23-s − 1.11e23·24-s − 7.58e22·25-s + 3.26e23·26-s − 4.41e23·27-s + 1.14e24·28-s + ⋯
L(s)  = 1  − 1.84·2-s + 0.716·3-s + 2.41·4-s − 0.590·5-s − 1.32·6-s + 0.625·7-s − 2.62·8-s − 0.486·9-s + 1.09·10-s − 0.537·11-s + 1.73·12-s − 0.793·13-s − 1.15·14-s − 0.423·15-s + 2.43·16-s − 1.66·17-s + 0.899·18-s − 0.111·19-s − 1.42·20-s + 0.448·21-s + 0.993·22-s + 1.06·23-s − 1.87·24-s − 0.651·25-s + 1.46·26-s − 1.06·27-s + 1.51·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(34-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+33/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(33\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :33/2),\ -1)\)
\(L(17)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{35}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 + 1.71e5T + 8.58e9T^{2} \)
3 \( 1 - 5.34e7T + 5.55e15T^{2} \)
5 \( 1 + 2.01e11T + 1.16e23T^{2} \)
7 \( 1 - 5.50e13T + 7.73e27T^{2} \)
11 \( 1 + 8.18e16T + 2.32e34T^{2} \)
13 \( 1 + 1.90e18T + 5.75e36T^{2} \)
17 \( 1 + 3.33e20T + 4.02e40T^{2} \)
19 \( 1 + 1.40e20T + 1.58e42T^{2} \)
23 \( 1 - 3.12e22T + 8.65e44T^{2} \)
29 \( 1 + 1.50e24T + 1.81e48T^{2} \)
31 \( 1 - 5.18e23T + 1.64e49T^{2} \)
37 \( 1 + 3.01e25T + 5.63e51T^{2} \)
41 \( 1 + 2.18e26T + 1.66e53T^{2} \)
43 \( 1 - 1.76e27T + 8.02e53T^{2} \)
47 \( 1 - 3.25e27T + 1.51e55T^{2} \)
53 \( 1 - 9.17e27T + 7.96e56T^{2} \)
59 \( 1 + 1.18e29T + 2.74e58T^{2} \)
61 \( 1 + 9.92e27T + 8.23e58T^{2} \)
67 \( 1 - 1.11e30T + 1.82e60T^{2} \)
71 \( 1 - 7.58e29T + 1.23e61T^{2} \)
73 \( 1 + 6.06e30T + 3.08e61T^{2} \)
79 \( 1 + 5.57e30T + 4.18e62T^{2} \)
83 \( 1 - 4.13e31T + 2.13e63T^{2} \)
89 \( 1 - 6.21e31T + 2.13e64T^{2} \)
97 \( 1 - 4.04e32T + 3.65e65T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.53206884006730080481924156153, −20.47185464506878550620095650151, −19.25851336377166860001110437885, −17.42233318430714526366111398424, −15.31150255031509647696477568593, −11.16259272438525703707451648647, −8.942750425824562441874080636099, −7.58479795202813569245751961691, −2.31107464957568549956618907631, 0, 2.31107464957568549956618907631, 7.58479795202813569245751961691, 8.942750425824562441874080636099, 11.16259272438525703707451648647, 15.31150255031509647696477568593, 17.42233318430714526366111398424, 19.25851336377166860001110437885, 20.47185464506878550620095650151, 24.53206884006730080481924156153

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