## Dirichlet series

L(s) = 1^{} | + 1.34·2^{-s} − 0.187·3^{-s} + 0.464·4^{-s} − 0.00162·5^{-s} − 0.251·6^{-s} + 0.228·7^{-s} + 0.169·8^{-s} − 0.463·9^{-s} − 0.00218·10^{-s} + 0.695·11^{-s} − 0.0870·12^{-s} − 0.882·13^{-s} + 0.306·14^{-s} + 0.000304·15^{-s} + 0.408·16^{-s} + 0.716·17^{-s} − 0.622·18^{-s} − 0.927·19^{-s} − 0.000699·20^{-s} − 0.0427·21^{-s} + 0.934·22^{-s} + 0.419·23^{-s} − 0.0309·24^{-s} + 0.227·25^{-s} − 1.17·26^{-s} + 0.100·28^{-s} + ⋯ |

## Functional equation

## Invariants

Degree: | \(4\) |

Conductor: | \(1\) |

Sign: | $1$ |

Analytic conductor: | \(2.21367\) |

Root analytic conductor: | \(1.21977\) |

Rational: | no |

Arithmetic: | no |

Primitive: | yes |

Self-dual: | yes |

Selberg data: | \((4,\ 1,\ (12.46875226152i, 4.72095103638i, -12.46875226152i, -4.72095103638i:\ ),\ 1)\) |

## Euler product

## Imaginary part of the first few zeros on the critical line

−24.342525613, −23.108966361, −22.396069285, −21.193386862, −19.439354578, −17.114451933, −14.496061510, 14.496061510, 17.114451933, 19.439354578, 21.193386862, 22.396069285, 23.108966361, 24.342525613

## 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.