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Properties

Label 1-311-311.113-r0-0-0
Degree $1$
Conductor $311$
Sign $0.986 - 0.162i$
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  + (−0.874 + 0.485i)2-s + (−0.874 − 0.485i)3-s + (0.528 − 0.848i)4-s + (−0.612 − 0.790i)5-s + 6-s + (0.979 + 0.201i)7-s + (−0.0506 + 0.998i)8-s + (0.528 + 0.848i)9-s + (0.918 + 0.394i)10-s + (−0.250 − 0.968i)11-s + (−0.874 + 0.485i)12-s + (0.528 + 0.848i)13-s + (−0.954 + 0.299i)14-s + (0.151 + 0.988i)15-s + (−0.440 − 0.897i)16-s + (0.918 + 0.394i)17-s + ⋯
L(s)  = 1  + (−0.874 + 0.485i)2-s + (−0.874 − 0.485i)3-s + (0.528 − 0.848i)4-s + (−0.612 − 0.790i)5-s + 6-s + (0.979 + 0.201i)7-s + (−0.0506 + 0.998i)8-s + (0.528 + 0.848i)9-s + (0.918 + 0.394i)10-s + (−0.250 − 0.968i)11-s + (−0.874 + 0.485i)12-s + (0.528 + 0.848i)13-s + (−0.954 + 0.299i)14-s + (0.151 + 0.988i)15-s + (−0.440 − 0.897i)16-s + (0.918 + 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.986 - 0.162i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.986 - 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6186060869 - 0.05062477132i\)
\(L(\frac12)\) \(\approx\) \(0.6186060869 - 0.05062477132i\)
\(L(1)\) \(\approx\) \(0.5928203038 + 0.02259234308i\)
\(L(1)\) \(\approx\) \(0.5928203038 + 0.02259234308i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.874 + 0.485i)T \)
3 \( 1 + (-0.874 - 0.485i)T \)
5 \( 1 + (-0.612 - 0.790i)T \)
7 \( 1 + (0.979 + 0.201i)T \)
11 \( 1 + (-0.250 - 0.968i)T \)
13 \( 1 + (0.528 + 0.848i)T \)
17 \( 1 + (0.918 + 0.394i)T \)
19 \( 1 + (-0.612 + 0.790i)T \)
23 \( 1 + (-0.0506 + 0.998i)T \)
29 \( 1 + (-0.954 - 0.299i)T \)
31 \( 1 + (0.918 - 0.394i)T \)
37 \( 1 + (0.979 - 0.201i)T \)
41 \( 1 + (0.347 - 0.937i)T \)
43 \( 1 + (0.979 + 0.201i)T \)
47 \( 1 + (-0.954 - 0.299i)T \)
53 \( 1 + (0.979 + 0.201i)T \)
59 \( 1 + (0.979 + 0.201i)T \)
61 \( 1 + (-0.612 + 0.790i)T \)
67 \( 1 + (0.347 + 0.937i)T \)
71 \( 1 + (0.820 - 0.571i)T \)
73 \( 1 + (-0.994 - 0.101i)T \)
79 \( 1 + (0.347 - 0.937i)T \)
83 \( 1 + (0.151 - 0.988i)T \)
89 \( 1 + (0.979 - 0.201i)T \)
97 \( 1 + (0.820 - 0.571i)T \)
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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

−25.63072261329425465758778584548, −24.34402799471203097336006199822, −23.21020498957877935363547751465, −22.63809179888528357941247019532, −21.51892719938497883705312818727, −20.74415563725029392908363027157, −19.98851952599735641309688161541, −18.63026734602465242230528724383, −18.05238120603108269818253920053, −17.39079621614327365573141073485, −16.346222452025573483217849189926, −15.385133220044074844555897692008, −14.70071720764379850040616418337, −12.85892111421441196862695584677, −11.94614275040896794151287477756, −11.08857644730511248393148966833, −10.58261124722109595683006364780, −9.69452412341754413787130273268, −8.24363659813419733616147152561, −7.42800855670797695712776520645, −6.437561561226324133085921916281, −4.86837627536814765896425610741, −3.849296971303681392139302328414, −2.549662404225435381371566116716, −0.89548523680608904583434996638, 0.93186175951574247925427953256, 1.84762862825308809445289266721, 4.17293402120177473550360099231, 5.466355377694454399380759046693, 6.01488991870284715523706761415, 7.536877001486298964466633474051, 8.072735738845896090805634622498, 9.00335530452555765464929613435, 10.430421806299514238294427679203, 11.44519504643825311023878236405, 11.81033783663406089921167355550, 13.22489995776212967128129890657, 14.392895989601484483611205412254, 15.5710425934905701474229082419, 16.49336233445674308401527413451, 16.920920644577130998869397058434, 17.93131369265207745906326425476, 18.92662187930523004842060879705, 19.27215457029683111709099186385, 20.82167236960674413949408675169, 21.36543130953501158659454369716, 23.08636551265585284348407455969, 23.73431710608568859934350826234, 24.25082264692730684742595006471, 24.962275296179604183685198361394

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