Properties

Label 1-968-968.261-r1-0-0
Degree $1$
Conductor $968$
Sign $0.990 + 0.137i$
Analytic cond. $104.026$
Root an. cond. $104.026$
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  + (0.809 − 0.587i)3-s + (−0.516 + 0.856i)5-s + (−0.198 + 0.980i)7-s + (0.309 − 0.951i)9-s + (−0.254 − 0.967i)13-s + (0.0855 + 0.996i)15-s + (0.985 − 0.170i)17-s + (0.696 + 0.717i)19-s + (0.415 + 0.909i)21-s + (0.415 − 0.909i)23-s + (−0.466 − 0.884i)25-s + (−0.309 − 0.951i)27-s + (−0.466 + 0.884i)29-s + (0.774 − 0.633i)31-s + (−0.736 − 0.676i)35-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.516 + 0.856i)5-s + (−0.198 + 0.980i)7-s + (0.309 − 0.951i)9-s + (−0.254 − 0.967i)13-s + (0.0855 + 0.996i)15-s + (0.985 − 0.170i)17-s + (0.696 + 0.717i)19-s + (0.415 + 0.909i)21-s + (0.415 − 0.909i)23-s + (−0.466 − 0.884i)25-s + (−0.309 − 0.951i)27-s + (−0.466 + 0.884i)29-s + (0.774 − 0.633i)31-s + (−0.736 − 0.676i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.990 + 0.137i$
Analytic conductor: \(104.026\)
Root analytic conductor: \(104.026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 968,\ (1:\ ),\ 0.990 + 0.137i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.641442253 + 0.1820277328i\)
\(L(\frac12)\) \(\approx\) \(2.641442253 + 0.1820277328i\)
\(L(1)\) \(\approx\) \(1.347658501 + 1.860211840\times10^{-5}i\)
\(L(1)\) \(\approx\) \(1.347658501 + 1.860211840\times10^{-5}i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.516 + 0.856i)T \)
7 \( 1 + (-0.198 + 0.980i)T \)
13 \( 1 + (-0.254 - 0.967i)T \)
17 \( 1 + (0.985 - 0.170i)T \)
19 \( 1 + (0.696 + 0.717i)T \)
23 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (-0.466 + 0.884i)T \)
31 \( 1 + (0.774 - 0.633i)T \)
37 \( 1 + (0.362 + 0.931i)T \)
41 \( 1 + (-0.610 + 0.791i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (0.941 - 0.336i)T \)
53 \( 1 + (-0.993 + 0.113i)T \)
59 \( 1 + (-0.610 - 0.791i)T \)
61 \( 1 + (-0.0285 - 0.999i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (0.870 - 0.491i)T \)
79 \( 1 + (0.921 - 0.389i)T \)
83 \( 1 + (-0.736 + 0.676i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.516 + 0.856i)T \)
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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

−21.19020848331111444377581596540, −20.83227378390741595771792268891, −19.86045141505060435971829523995, −19.51741463405725542395771378634, −18.712987675119758324544001873847, −17.16330595972812523759932886491, −16.83087764655801208551440143757, −15.8910257954488560883185267907, −15.399185209157887637397194334539, −14.20471997185810576251366042588, −13.73857440055345168859548996416, −12.90182269132087521485017435285, −11.87439667957146195017254813833, −11.04250143973317979145864012488, −9.95751557214468284171887217263, −9.39685155898311954925793502166, −8.573495700093258584364288797849, −7.606886052250154304848880998031, −7.118937796214302909815607849221, −5.50578824684730596682189344982, −4.61719302702100204820019345542, −3.92332473453983650498967954380, −3.17435878165325308618186932506, −1.77914408576260996967080262521, −0.71055883275277742553104820006, 0.77147335041732715187726034348, 2.09782413967128758419796412499, 3.11380282392287639116300571461, 3.34920749703286670163773142270, 4.93986253249132872912723063417, 6.07487924132291900361537384691, 6.79723921549013319061752493108, 7.99319927426958883303505874292, 8.0756279072574627519567072323, 9.44294893403886540606724204478, 10.062360434833762516932443679968, 11.22852005428970525247990213612, 12.20325934885929286358227495486, 12.583448637002024723919111966, 13.71691395697889691916285821207, 14.57818684446422030872940834177, 15.05424498850577692738008500158, 15.73155105276462507108209420483, 16.83652286617161877608545614477, 18.10432103598643543969595450273, 18.57902823398444162969287488039, 18.97108018291378332931441449546, 20.00142674160249250331171571124, 20.557820715780118075184184372978, 21.60019311622005300846656400388

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