Properties

Degree 1
Conductor 311
Sign $0.0717 - 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.820 − 0.571i)2-s + (0.820 + 0.571i)3-s + (0.347 − 0.937i)4-s + (0.151 − 0.988i)5-s + 6-s + (0.528 − 0.848i)7-s + (−0.250 − 0.968i)8-s + (0.347 + 0.937i)9-s + (−0.440 − 0.897i)10-s + (0.954 − 0.299i)11-s + (0.820 − 0.571i)12-s + (0.347 + 0.937i)13-s + (−0.0506 − 0.998i)14-s + (0.688 − 0.724i)15-s + (−0.758 − 0.651i)16-s + (0.440 + 0.897i)17-s + ⋯
L(s,χ)  = 1  + (0.820 − 0.571i)2-s + (0.820 + 0.571i)3-s + (0.347 − 0.937i)4-s + (0.151 − 0.988i)5-s + 6-s + (0.528 − 0.848i)7-s + (−0.250 − 0.968i)8-s + (0.347 + 0.937i)9-s + (−0.440 − 0.897i)10-s + (0.954 − 0.299i)11-s + (0.820 − 0.571i)12-s + (0.347 + 0.937i)13-s + (−0.0506 − 0.998i)14-s + (0.688 − 0.724i)15-s + (−0.758 − 0.651i)16-s + (0.440 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0717 - 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0717 - 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $0.0717 - 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (264, \cdot )$
Sato-Tate  :  $\mu(62)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 311,\ (1:\ ),\ 0.0717 - 0.997i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.672279936 - 3.417531229i$
$L(\frac12,\chi)$  $\approx$  $3.672279936 - 3.417531229i$
$L(\chi,1)$  $\approx$  2.257985787 - 1.123917403i
$L(1,\chi)$  $\approx$  2.257985787 - 1.123917403i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−25.1693626679383068349319076632, −24.65947350355750196803977040516, −23.39642685661899891981130471110, −22.65112120834445265951921446236, −21.81654915077639846323033193916, −20.828003384569129264110420992068, −20.07477766666002293764647261879, −18.653651316626687116586607817282, −18.19254258601084760005599207452, −17.12487480255468198385672761002, −15.677365527260628100718982184697, −14.87720927096158042695566018182, −14.39397438304425269116485033572, −13.61366272993557434404764641271, −12.33117552508145793046229699896, −11.84789927463652551792724416337, −10.37757148459818686107181789857, −8.89037778746710841697902928459, −8.09620904628292048755388588081, −7.04647595503475944398245936995, −6.33019724574148414875845668769, −5.15745827127928320571871213319, −3.59491896284909197097606670269, −2.87594794853602944123352272136, −1.743135387613492609984600609951, 1.10757220143199955686277957985, 1.99544546254899290035000645662, 3.68120853170574545871446886207, 4.21009974952322649375246763136, 5.12286208199382277812113766190, 6.50296584369555135666315229343, 7.94358610915674077343728689130, 9.09598504358595156657692139284, 9.79071550546630259854290665240, 11.00229239897082982131275427873, 11.78943389508404463793715846733, 13.191817010974858018176042905238, 13.6880640508235073698314195607, 14.507002415301420436126138096984, 15.461587837465303824626232976600, 16.507166271876683138474553968712, 17.30985232755510435896430210498, 19.14895156430605067868071670447, 19.65847223147619070747964006070, 20.46293229097492059389821769628, 21.31681942812197013074063713096, 21.62221736721257665145386433400, 23.019910637846255953586173160777, 24.03484342921273480518690109120, 24.50251165366736333237239219766

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