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Properties

Label 1-311-311.264-r1-0-0
Degree $1$
Conductor $311$
Sign $0.0717 - 0.997i$
Analytic cond. $33.4215$
Root an. cond. $33.4215$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.0717 - 0.997i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.0717 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.672279936 - 3.417531229i\)
\(L(\frac12)\) \(\approx\) \(3.672279936 - 3.417531229i\)
\(L(1)\) \(\approx\) \(2.257985787 - 1.123917403i\)
\(L(1)\) \(\approx\) \(2.257985787 - 1.123917403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.820 - 0.571i)T \)
3 \( 1 + (0.820 + 0.571i)T \)
5 \( 1 + (0.151 - 0.988i)T \)
7 \( 1 + (0.528 - 0.848i)T \)
11 \( 1 + (0.954 - 0.299i)T \)
13 \( 1 + (0.347 + 0.937i)T \)
17 \( 1 + (0.440 + 0.897i)T \)
19 \( 1 + (-0.151 - 0.988i)T \)
23 \( 1 + (0.250 + 0.968i)T \)
29 \( 1 + (0.0506 - 0.998i)T \)
31 \( 1 + (0.440 - 0.897i)T \)
37 \( 1 + (-0.528 - 0.848i)T \)
41 \( 1 + (-0.979 + 0.201i)T \)
43 \( 1 + (-0.528 + 0.848i)T \)
47 \( 1 + (-0.0506 + 0.998i)T \)
53 \( 1 + (0.528 - 0.848i)T \)
59 \( 1 + (-0.528 + 0.848i)T \)
61 \( 1 + (-0.151 - 0.988i)T \)
67 \( 1 + (0.979 + 0.201i)T \)
71 \( 1 + (0.994 - 0.101i)T \)
73 \( 1 + (-0.874 + 0.485i)T \)
79 \( 1 + (0.979 - 0.201i)T \)
83 \( 1 + (0.688 + 0.724i)T \)
89 \( 1 + (0.528 + 0.848i)T \)
97 \( 1 + (0.994 - 0.101i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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