Properties

Degree 2
Conductor 5
Sign $-1$
Motivic weight 33
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.61e4·2-s + 5.48e7·3-s − 5.44e9·4-s − 1.52e11·5-s − 3.07e12·6-s + 7.38e13·7-s + 7.87e14·8-s − 2.54e15·9-s + 8.56e15·10-s + 3.54e16·11-s − 2.98e17·12-s − 1.27e18·13-s − 4.14e18·14-s − 8.37e18·15-s + 2.54e18·16-s + 1.95e20·17-s + 1.43e20·18-s + 9.67e20·19-s + 8.30e20·20-s + 4.05e21·21-s − 1.98e21·22-s − 3.06e22·23-s + 4.31e22·24-s + 2.32e22·25-s + 7.14e22·26-s − 4.44e23·27-s − 4.01e23·28-s + ⋯
L(s)  = 1  − 0.605·2-s + 0.735·3-s − 0.633·4-s − 0.447·5-s − 0.445·6-s + 0.839·7-s + 0.989·8-s − 0.458·9-s + 0.270·10-s + 0.232·11-s − 0.466·12-s − 0.530·13-s − 0.508·14-s − 0.329·15-s + 0.0345·16-s + 0.974·17-s + 0.277·18-s + 0.769·19-s + 0.283·20-s + 0.618·21-s − 0.140·22-s − 1.04·23-s + 0.727·24-s + 0.200·25-s + 0.321·26-s − 1.07·27-s − 0.532·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(33\)
character  :  $\chi_{5} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 5,\ (\ :33/2),\ -1)$
$L(17)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{35}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + 1.52e11T \)
good2 \( 1 + 5.61e4T + 8.58e9T^{2} \)
3 \( 1 - 5.48e7T + 5.55e15T^{2} \)
7 \( 1 - 7.38e13T + 7.73e27T^{2} \)
11 \( 1 - 3.54e16T + 2.32e34T^{2} \)
13 \( 1 + 1.27e18T + 5.75e36T^{2} \)
17 \( 1 - 1.95e20T + 4.02e40T^{2} \)
19 \( 1 - 9.67e20T + 1.58e42T^{2} \)
23 \( 1 + 3.06e22T + 8.65e44T^{2} \)
29 \( 1 + 1.16e23T + 1.81e48T^{2} \)
31 \( 1 + 3.89e23T + 1.64e49T^{2} \)
37 \( 1 + 2.73e25T + 5.63e51T^{2} \)
41 \( 1 + 7.27e26T + 1.66e53T^{2} \)
43 \( 1 + 1.74e27T + 8.02e53T^{2} \)
47 \( 1 - 2.14e26T + 1.51e55T^{2} \)
53 \( 1 + 1.96e28T + 7.96e56T^{2} \)
59 \( 1 - 2.54e29T + 2.74e58T^{2} \)
61 \( 1 + 2.81e29T + 8.23e58T^{2} \)
67 \( 1 + 8.18e29T + 1.82e60T^{2} \)
71 \( 1 + 3.59e30T + 1.23e61T^{2} \)
73 \( 1 + 2.19e30T + 3.08e61T^{2} \)
79 \( 1 - 3.90e31T + 4.18e62T^{2} \)
83 \( 1 - 4.98e30T + 2.13e63T^{2} \)
89 \( 1 + 1.23e32T + 2.13e64T^{2} \)
97 \( 1 - 1.97e32T + 3.65e65T^{2} \)
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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

−14.76858011878060215338185351988, −13.80114750990936327619209505245, −11.78744284816231398155067914872, −9.931287190093027734145427264409, −8.527039804162388425733647181615, −7.69301618843731490330940734945, −5.06693215451136000491998908408, −3.48760672775751725715664946117, −1.59830898057801865036827199718, 0, 1.59830898057801865036827199718, 3.48760672775751725715664946117, 5.06693215451136000491998908408, 7.69301618843731490330940734945, 8.527039804162388425733647181615, 9.931287190093027734145427264409, 11.78744284816231398155067914872, 13.80114750990936327619209505245, 14.76858011878060215338185351988

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