Properties

Degree 4
Conductor 1
Sign $1$
Self-dual yes

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Dirichlet series

$L(s,f)$  = 1  + 1.342·2-s − 0.187·3-s + 0.464·4-s − 0.001·5-s − 0.251·6-s + 0.228·7-s + 0.169·8-s − 0.463·9-s − 0.002·10-s + 0.695·11-s − 0.087·12-s − 0.882·13-s + 0.306·14-s + 0.000·15-s + 0.408·16-s + 0.716·17-s − 0.622·18-s − 0.927·19-s − 0.000·20-s − 0.042·21-s + 0.934·22-s + 0.42·23-s − 0.031·24-s + 0.227·25-s − 1.18·26-s + 0.1·28-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+12.46i)\Gamma_{\R}(s+4.720i)\Gamma_{\R}(s-12.46 i)\Gamma_{\R}(s-4.720 i) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,f) \end{align} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 1,\ (12.4687522615i, 4.72095103638i, -12.4687522615i, -4.72095103638i:\ ),\ 1)$

Euler product

\[\begin{equation} L(s,f) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

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

The degree 4 L-function with conductor 1 and spectral parameters approximately $\pm 4.7209 i, \pm 12.4687 i$ has the surprising property that its first nontrivial zeros have imaginary part $\pm 14.496\ldots$.

This is surprising because the Riemann zeta function has its first zeros with imaginary part $\pm 14.134\ldots$, which is a gap of $28.269\ldots$. It had been proven [MR:1890648] that the Riemann zeta function has the largest gap among L-functions with real spectral parameters. It had been (mistakenly) thought that the zeta function should have the largest gap among all L-functions, but this example illustrates how the trivial zeros, which come from the spectral parameters, can create a larger gap between the nontrivial zeros.