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Properties

Label 1-311-311.47-r0-0-0
Degree $1$
Conductor $311$
Sign $0.335 - 0.942i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.335 - 0.942i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.335 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.147783208 - 1.515659177i\)
\(L(\frac12)\) \(\approx\) \(2.147783208 - 1.515659177i\)
\(L(1)\) \(\approx\) \(1.903963796 - 0.8195696933i\)
\(L(1)\) \(\approx\) \(1.903963796 - 0.8195696933i\)

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

−25.318307250559564761903048386, −24.53992900344014555763587194800, −23.75839001219569318398893218781, −22.88950127842405775144351563196, −21.70864167535609573656091805022, −21.294836305651444349304265899656, −20.14851099090435575702860836618, −19.06384810651538194669217380979, −17.97391012345584139596973058289, −17.66303087254903650038502562776, −15.70936961533490820441726148658, −15.233506580553678227887127631, −14.5729978180506923551910845277, −13.428055240767027511316059391407, −13.01564511736500461716436446465, −11.709394606877824638573147906977, −10.8025458713825708989308399926, −9.19919093918065416258527898970, −8.00343643643257322181107898458, −7.58772685799130297036317008698, −6.24038653207978065604475348448, −5.56575396642185764084449879774, −3.9357396107849557741327687265, −2.80534254383040509889770983557, −2.22458803333496172283309980624, 1.390339496921960078519999254696, 2.45123395910553069513032302963, 3.874680539283172957243114285443, 4.58446222648653360765285522906, 5.33874960017916769989473460535, 7.036473393325897580134291295938, 8.24065779342826345071417036705, 9.311688303200023603447368744, 10.23296469215956097999317291649, 11.06585242623671422363834518338, 12.33859708176354973894165433711, 13.28425274719865312091136842188, 13.98827752048259640417188099754, 14.6754674835115627197330847055, 16.10275603961749356358374715875, 16.31433631550156537328290497588, 18.02926424909886950242840922674, 19.124460749700535962049918582006, 20.27612092226261766609103485532, 20.59936853778726383022464700593, 21.131408427862919393531891290415, 22.18895844044946753616558823410, 23.37630128318002743011814555691, 24.06205250158728653726686322733, 24.877682587822726386595479633698

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