## Dirichlet series

L(s) = 1^{} | + (−0.679 + 0.434i)2^{-s} + (0.572 − 0.610i)3^{-s} + (−0.128 − 0.591i)4^{-s} + (−0.117 − 0.0119i)5^{-s} + (−0.124 + 0.663i)6^{-s} + (0.953 − 0.364i)7^{-s} + (−0.0628 − 0.263i)8^{-s} + (−0.241 − 0.699i)9^{-s} + (0.0850 − 0.0428i)10^{-s} + (−0.524 − 0.247i)11^{-s} + (−0.434 − 0.260i)12^{-s} + (−0.317 + 0.138i)13^{-s} + (−0.489 + 0.662i)14^{-s} + (−0.0745 + 0.0647i)15^{-s} + (−0.140 + 0.388i)16^{-s} + (0.596 + 0.309i)17^{-s} + ⋯ |

## Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+23.5i) \, \Gamma_{\R}(s-0.822i) \, \Gamma_{\R}(s-8.12i) \, \Gamma_{\R}(s-14.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]

## Invariants

Degree: | \(4\) |

Conductor: | \(1\) |

Sign: | $1$ |

Analytic conductor: | \(1.07364\) |

Root analytic conductor: | \(1.01792\) |

Rational: | no |

Arithmetic: | no |

Primitive: | yes |

Self-dual: | no |

Selberg data: | \((4,\ 1,\ (23.5430066352i, -0.822180974528i, -8.12585455838i, -14.59497110224i:\ ),\ 1)\) |

## Euler product

\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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

−21.37532528, −20.26415920, −18.54570547, −17.26088332, −15.43215124, −14.07956742, −11.99956019, −10.51519076, −8.97047993, −7.97719801, −4.86505332, −2.60944945, 17.83304504, 19.27632994, 21.05750458, 23.44783889, 24.38990213

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

The first positive critical zero of this L-function, at height approximately 17.833, is higher than any other L-function of conductor 1 and signature (0,0,0,0;). The first negative zero is at height approximately −2.609 and there are several other degree 4 L-functions with a larger gap between zeros.