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Properties

Label 1-311-311.293-r1-0-0
Degree $1$
Conductor $311$
Sign $0.111 + 0.993i$
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  + (0.528 − 0.848i)2-s + (0.528 + 0.848i)3-s + (−0.440 − 0.897i)4-s + (−0.250 + 0.968i)5-s + 6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (−0.440 + 0.897i)9-s + (0.688 + 0.724i)10-s + (0.874 − 0.485i)11-s + (0.528 − 0.848i)12-s + (−0.440 + 0.897i)13-s + (0.820 − 0.571i)14-s + (−0.954 + 0.299i)15-s + (−0.612 + 0.790i)16-s + (−0.688 − 0.724i)17-s + ⋯
L(s)  = 1  + (0.528 − 0.848i)2-s + (0.528 + 0.848i)3-s + (−0.440 − 0.897i)4-s + (−0.250 + 0.968i)5-s + 6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (−0.440 + 0.897i)9-s + (0.688 + 0.724i)10-s + (0.874 − 0.485i)11-s + (0.528 − 0.848i)12-s + (−0.440 + 0.897i)13-s + (0.820 − 0.571i)14-s + (−0.954 + 0.299i)15-s + (−0.612 + 0.790i)16-s + (−0.688 − 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.111 + 0.993i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.111 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.814562222 + 1.623139459i\)
\(L(\frac12)\) \(\approx\) \(1.814562222 + 1.623139459i\)
\(L(1)\) \(\approx\) \(1.502378721 + 0.2159610677i\)
\(L(1)\) \(\approx\) \(1.502378721 + 0.2159610677i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.528 - 0.848i)T \)
3 \( 1 + (0.528 + 0.848i)T \)
5 \( 1 + (-0.250 + 0.968i)T \)
7 \( 1 + (0.918 + 0.394i)T \)
11 \( 1 + (0.874 - 0.485i)T \)
13 \( 1 + (-0.440 + 0.897i)T \)
17 \( 1 + (-0.688 - 0.724i)T \)
19 \( 1 + (0.250 + 0.968i)T \)
23 \( 1 + (0.994 + 0.101i)T \)
29 \( 1 + (-0.820 - 0.571i)T \)
31 \( 1 + (-0.688 + 0.724i)T \)
37 \( 1 + (-0.918 + 0.394i)T \)
41 \( 1 + (0.758 + 0.651i)T \)
43 \( 1 + (-0.918 - 0.394i)T \)
47 \( 1 + (0.820 + 0.571i)T \)
53 \( 1 + (0.918 + 0.394i)T \)
59 \( 1 + (-0.918 - 0.394i)T \)
61 \( 1 + (0.250 + 0.968i)T \)
67 \( 1 + (-0.758 + 0.651i)T \)
71 \( 1 + (-0.347 + 0.937i)T \)
73 \( 1 + (0.979 + 0.201i)T \)
79 \( 1 + (-0.758 - 0.651i)T \)
83 \( 1 + (-0.954 - 0.299i)T \)
89 \( 1 + (0.918 - 0.394i)T \)
97 \( 1 + (-0.347 + 0.937i)T \)
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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

−24.472444408227248934936348403790, −24.29519274708312144382729785967, −23.36515926913476395901631157124, −22.39453614646495373592358439498, −21.17160423249795399273272848032, −20.25419229714586555111807275272, −19.69793857292719543414407852979, −18.16544942010678457537897130364, −17.33290307628592189676759524128, −16.928115936374096531483014510314, −15.29875922000161327714792938790, −14.8675813840692841741190122107, −13.76100150141371810244325826127, −12.96053158885630516415455363672, −12.304695672605606785476322876493, −11.25084663623300576664944726847, −9.19856254515774479253881300932, −8.610112264643099446530227437252, −7.59755580445603947319384520918, −6.99264021207980083041530901547, −5.57917818983803271255739611758, −4.60218467496045686789290768238, −3.573186446434399476930019761898, −1.9342403308134976722641325412, −0.54511246795789153135133307230, 1.73336100787848026469550272276, 2.77250880787584977947531650885, 3.7750583501833145124741732070, 4.63224084679118904730808104072, 5.76624136980669560326355283894, 7.19313108823771141510111380046, 8.693188663252170253026021985, 9.41962506441641744163984076888, 10.5456203296515843124003078714, 11.36425807538214775193851506413, 11.86414593301300296351256889855, 13.60816606668047272623613764522, 14.398429272013038598464271539081, 14.75522616898718508412601608154, 15.76108490859953025264680366534, 17.10774146113113773966643016353, 18.44398741619295923283326639075, 19.08104794619720501192801560245, 19.97023013925481353186495003855, 20.919617202802256176086907993297, 21.6596895065324617038145927992, 22.25165280537161460769717612554, 23.06331822133797913401918650611, 24.333993058932923970298844621702, 25.084659767122532365672221808756

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