# Properties

 Label 4-1-1.1-r0e4-m0.82m8.13m14.59p23.54-0 Degree $4$ Conductor $1$ Sign $1$ Analytic cond. $1.07364$ Root an. cond. $1.01792$ Arithmetic no Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## 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.