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  + 7.19e4·2-s + 1.37e8·3-s − 3.41e9·4-s − 1.52e11·5-s + 9.89e12·6-s − 1.07e14·7-s − 8.63e14·8-s + 1.33e16·9-s − 1.09e16·10-s − 2.33e17·11-s − 4.69e17·12-s − 2.28e18·13-s − 7.74e18·14-s − 2.09e19·15-s − 3.27e19·16-s − 2.09e20·17-s + 9.61e20·18-s + 6.01e20·19-s + 5.21e20·20-s − 1.48e22·21-s − 1.67e22·22-s + 4.84e22·23-s − 1.18e23·24-s + 2.32e22·25-s − 1.64e23·26-s + 1.07e24·27-s + 3.67e23·28-s + ⋯
L(s)  = 1  + 0.776·2-s + 1.84·3-s − 0.397·4-s − 0.447·5-s + 1.43·6-s − 1.22·7-s − 1.08·8-s + 2.40·9-s − 0.347·10-s − 1.53·11-s − 0.733·12-s − 0.951·13-s − 0.950·14-s − 0.825·15-s − 0.444·16-s − 1.04·17-s + 1.86·18-s + 0.478·19-s + 0.177·20-s − 2.25·21-s − 1.18·22-s + 1.64·23-s − 2.00·24-s + 0.200·25-s − 0.738·26-s + 2.59·27-s + 0.486·28-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 \( 1 - 7.19e4T + 8.58e9T^{2} \)
3 \( 1 - 1.37e8T + 5.55e15T^{2} \)
7 \( 1 + 1.07e14T + 7.73e27T^{2} \)
11 \( 1 + 2.33e17T + 2.32e34T^{2} \)
13 \( 1 + 2.28e18T + 5.75e36T^{2} \)
17 \( 1 + 2.09e20T + 4.02e40T^{2} \)
19 \( 1 - 6.01e20T + 1.58e42T^{2} \)
23 \( 1 - 4.84e22T + 8.65e44T^{2} \)
29 \( 1 + 8.15e22T + 1.81e48T^{2} \)
31 \( 1 + 1.49e24T + 1.64e49T^{2} \)
37 \( 1 - 1.78e25T + 5.63e51T^{2} \)
41 \( 1 + 1.94e26T + 1.66e53T^{2} \)
43 \( 1 + 9.48e26T + 8.02e53T^{2} \)
47 \( 1 + 6.29e26T + 1.51e55T^{2} \)
53 \( 1 + 8.56e27T + 7.96e56T^{2} \)
59 \( 1 - 5.43e27T + 2.74e58T^{2} \)
61 \( 1 - 1.79e29T + 8.23e58T^{2} \)
67 \( 1 + 3.33e29T + 1.82e60T^{2} \)
71 \( 1 + 3.44e30T + 1.23e61T^{2} \)
73 \( 1 + 7.11e30T + 3.08e61T^{2} \)
79 \( 1 + 1.57e31T + 4.18e62T^{2} \)
83 \( 1 - 3.45e31T + 2.13e63T^{2} \)
89 \( 1 + 2.35e32T + 2.13e64T^{2} \)
97 \( 1 + 3.55e32T + 3.65e65T^{2} \)
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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

−14.85604707098638048606948797079, −13.34331483242211711687889608657, −12.86438683136340060378063384481, −9.836396738468180996796894734876, −8.705838615392936199684050655286, −7.20665897752049466098142899409, −4.76194412056382722165578677778, −3.30689421011607447182734749356, −2.65789535174190475688200019494, 0, 2.65789535174190475688200019494, 3.30689421011607447182734749356, 4.76194412056382722165578677778, 7.20665897752049466098142899409, 8.705838615392936199684050655286, 9.836396738468180996796894734876, 12.86438683136340060378063384481, 13.34331483242211711687889608657, 14.85604707098638048606948797079

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