Properties

Label 4-1-1.1-r0e4-m0.88p8.89p14.51m22.52-0
Degree 44
Conductor 11
Sign 11
Analytic cond. 1.251301.25130
Root an. cond. 1.057641.05764
Arithmetic no
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Dirichlet series

L(s)  = 1  + (−0.473 + 0.345i)2-s + (0.902 + 0.0126i)3-s + (−0.525 − 0.327i)4-s + (−0.0303 − 0.847i)5-s + (−0.431 + 0.306i)6-s + (−0.348 − 0.611i)7-s + (0.187 − 0.590i)8-s + (0.799 + 0.0227i)9-s + (0.307 + 0.391i)10-s + (0.256 + 0.745i)11-s + (−0.470 − 0.302i)12-s + (−0.477 + 0.472i)13-s + (0.376 + 0.169i)14-s + (−0.0167 − 0.765i)15-s + (−0.209 + 0.551i)16-s + (0.205 − 0.140i)17-s + ⋯

Functional equation

Λ(s)=(ΓR(s22.5i)ΓR(s0.877i)ΓR(s+8.89i)ΓR(s+14.5i)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-22.5i) \, \Gamma_{\R}(s-0.877i) \, \Gamma_{\R}(s+8.89i) \, \Gamma_{\R}(s+14.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}

Invariants

Degree: 44
Conductor: 11
Sign: 11
Analytic conductor: 1.251301.25130
Root analytic conductor: 1.057641.05764
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: no
Selberg data: (4, 1, (22.5224000508i,0.877793680384i,8.89271031532i,14.50748341588i: ), 1)(4,\ 1,\ (-22.5224000508i, -0.877793680384i, 8.89271031532i, 14.50748341588i:\ ),\ 1)

Euler product

L(s)=p j=14(1αj,pps)1L(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

−24.67850357, −22.71078844, −21.38370873, −19.52958908, −18.29701111, −2.90315745, 4.36082312, 7.21584682, 8.92106206, 9.81023223, 12.66475243, 13.96602377, 15.62993783, 17.17431380, 18.94507322, 20.35447110, 24.40107476

Graph of the ZZ-function along the critical line