Properties

Label 4-1-1.1-r0e4-c4.72c12.47-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $2.21367$
Root an. cond. $1.21977$
Arithmetic no
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

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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

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+12.4i) \, \Gamma_{\R}(s+4.72i) \, \Gamma_{\R}(s-12.4i) \, \Gamma_{\R}(s-4.72i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

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

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

Imaginary part of the first few zeros on the critical line

−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.