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Properties

Label 1-2667-2667.131-r0-0-0
Degree $1$
Conductor $2667$
Sign $-0.308 + 0.951i$
Analytic cond. $12.3854$
Root an. cond. $12.3854$
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  + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (0.826 + 0.563i)5-s + (−0.623 + 0.781i)8-s + (−0.955 − 0.294i)10-s + (−0.955 − 0.294i)11-s + (0.900 − 0.433i)13-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + 20-s + 22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.733 + 0.680i)26-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (0.826 + 0.563i)5-s + (−0.623 + 0.781i)8-s + (−0.955 − 0.294i)10-s + (−0.955 − 0.294i)11-s + (0.900 − 0.433i)13-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + 20-s + 22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.733 + 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.308 + 0.951i$
Analytic conductor: \(12.3854\)
Root analytic conductor: \(12.3854\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (0:\ ),\ -0.308 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5145993091 + 0.7081941162i\)
\(L(\frac12)\) \(\approx\) \(0.5145993091 + 0.7081941162i\)
\(L(1)\) \(\approx\) \(0.7186685695 + 0.2001628585i\)
\(L(1)\) \(\approx\) \(0.7186685695 + 0.2001628585i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.955 + 0.294i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.955 + 0.294i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (-0.826 + 0.563i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (-0.988 + 0.149i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (0.955 - 0.294i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−18.88160659465738026577831549076, −18.34610241596980858900736917177, −17.80826250213840193329501362289, −17.05354627440951141737321302892, −16.47565431553196068921931636282, −15.76748060214490312053305614762, −15.11945269711280961115727876075, −13.96458415116588781441371429641, −13.218928301460784645860905906093, −12.68850475645039820964009618767, −11.867254930657576094478259364216, −10.98038938348005590974129250431, −10.34160215958878250928953996408, −9.76782973570626898930796353524, −8.84026673605052304894689184570, −8.504505629083777015322853792196, −7.55551448119778622639899664302, −6.72306863980140662846555888650, −5.91571616747938785809508198355, −5.179579055339936838405357959201, −4.02757043908571767696657122102, −3.14016400748557045442569040283, −1.987835289246345296798061070968, −1.69650791255091749875523248799, −0.39087831306712139753473044401, 1.05586406091961210930061743328, 1.95376225203987822682373740771, 2.77639984083488043185029921185, 3.53263357521616008732858124400, 5.07824441013771261929094687077, 5.884470287510152520915440818745, 6.12585666984901489181736332151, 7.479589135369151665783988568100, 7.62748226369417295804634747888, 8.752913550039242773553278850005, 9.40082179073824618323626455616, 10.19684207130570041587078819384, 10.63475550637395279192459271791, 11.364081950545031950853803230902, 12.22914307755330454430901019420, 13.35875046244151493657550434710, 13.880618182363973846317745204786, 14.64944070763302394768337292936, 15.47620343205396271621963912139, 16.13939080245878359893153702095, 16.631099562609119059636865905, 17.71729368731991459007155716958, 18.174927329811815541417131615292, 18.460597603623934711762329848901, 19.30778978823767336181675925646

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