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  + 5.62e3·2-s − 1.24e8·3-s − 8.55e9·4-s − 1.52e11·5-s − 7.01e11·6-s + 7.01e13·7-s − 9.65e13·8-s + 9.96e15·9-s − 8.58e14·10-s + 9.39e16·11-s + 1.06e18·12-s + 9.95e17·13-s + 3.94e17·14-s + 1.90e19·15-s + 7.29e19·16-s − 2.59e20·17-s + 5.60e19·18-s − 1.31e21·19-s + 1.30e21·20-s − 8.74e21·21-s + 5.28e20·22-s + 5.44e22·23-s + 1.20e22·24-s + 2.32e22·25-s + 5.60e21·26-s − 5.49e23·27-s − 6.00e23·28-s + ⋯
L(s)  = 1  + 0.0607·2-s − 1.67·3-s − 0.996·4-s − 0.447·5-s − 0.101·6-s + 0.797·7-s − 0.121·8-s + 1.79·9-s − 0.0271·10-s + 0.616·11-s + 1.66·12-s + 0.414·13-s + 0.0484·14-s + 0.747·15-s + 0.988·16-s − 1.29·17-s + 0.108·18-s − 1.04·19-s + 0.445·20-s − 1.33·21-s + 0.0374·22-s + 1.85·23-s + 0.202·24-s + 0.200·25-s + 0.0251·26-s − 1.32·27-s − 0.794·28-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 \( 1 - 5.62e3T + 8.58e9T^{2} \)
3 \( 1 + 1.24e8T + 5.55e15T^{2} \)
7 \( 1 - 7.01e13T + 7.73e27T^{2} \)
11 \( 1 - 9.39e16T + 2.32e34T^{2} \)
13 \( 1 - 9.95e17T + 5.75e36T^{2} \)
17 \( 1 + 2.59e20T + 4.02e40T^{2} \)
19 \( 1 + 1.31e21T + 1.58e42T^{2} \)
23 \( 1 - 5.44e22T + 8.65e44T^{2} \)
29 \( 1 + 9.77e23T + 1.81e48T^{2} \)
31 \( 1 - 6.82e24T + 1.64e49T^{2} \)
37 \( 1 + 7.50e25T + 5.63e51T^{2} \)
41 \( 1 + 2.94e25T + 1.66e53T^{2} \)
43 \( 1 + 1.00e27T + 8.02e53T^{2} \)
47 \( 1 - 1.91e27T + 1.51e55T^{2} \)
53 \( 1 - 4.33e28T + 7.96e56T^{2} \)
59 \( 1 + 1.21e29T + 2.74e58T^{2} \)
61 \( 1 - 4.03e29T + 8.23e58T^{2} \)
67 \( 1 + 1.33e30T + 1.82e60T^{2} \)
71 \( 1 + 4.10e30T + 1.23e61T^{2} \)
73 \( 1 - 8.64e30T + 3.08e61T^{2} \)
79 \( 1 + 1.76e31T + 4.18e62T^{2} \)
83 \( 1 + 6.43e30T + 2.13e63T^{2} \)
89 \( 1 - 1.32e32T + 2.13e64T^{2} \)
97 \( 1 - 2.90e32T + 3.65e65T^{2} \)
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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

−15.13750264624874768371020073832, −13.18044754966314464041737505683, −11.75705240373532426038355083913, −10.69383533862968044208426783688, −8.701726712481703178513907096448, −6.65713411629284494211826183269, −5.09588363391465376274668579748, −4.22078326040349388487576387208, −1.15229297651253518036906482209, 0, 1.15229297651253518036906482209, 4.22078326040349388487576387208, 5.09588363391465376274668579748, 6.65713411629284494211826183269, 8.701726712481703178513907096448, 10.69383533862968044208426783688, 11.75705240373532426038355083913, 13.18044754966314464041737505683, 15.13750264624874768371020073832

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