Properties

Label 1-968-968.101-r1-0-0
Degree $1$
Conductor $968$
Sign $0.999 + 0.0181i$
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  + (−0.309 + 0.951i)3-s + (−0.696 − 0.717i)5-s + (−0.516 + 0.856i)7-s + (−0.809 − 0.587i)9-s + (0.198 − 0.980i)13-s + (0.897 − 0.441i)15-s + (−0.610 + 0.791i)17-s + (−0.564 + 0.825i)19-s + (−0.654 − 0.755i)21-s + (−0.654 + 0.755i)23-s + (−0.0285 + 0.999i)25-s + (0.809 − 0.587i)27-s + (−0.0285 − 0.999i)29-s + (−0.870 + 0.491i)31-s + (0.974 − 0.226i)35-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (−0.696 − 0.717i)5-s + (−0.516 + 0.856i)7-s + (−0.809 − 0.587i)9-s + (0.198 − 0.980i)13-s + (0.897 − 0.441i)15-s + (−0.610 + 0.791i)17-s + (−0.564 + 0.825i)19-s + (−0.654 − 0.755i)21-s + (−0.654 + 0.755i)23-s + (−0.0285 + 0.999i)25-s + (0.809 − 0.587i)27-s + (−0.0285 − 0.999i)29-s + (−0.870 + 0.491i)31-s + (0.974 − 0.226i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.999 + 0.0181i$
Analytic conductor: \(104.026\)
Root analytic conductor: \(104.026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 968,\ (1:\ ),\ 0.999 + 0.0181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4843688517 + 0.004401702725i\)
\(L(\frac12)\) \(\approx\) \(0.4843688517 + 0.004401702725i\)
\(L(1)\) \(\approx\) \(0.5916897413 + 0.1697652235i\)
\(L(1)\) \(\approx\) \(0.5916897413 + 0.1697652235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.696 - 0.717i)T \)
7 \( 1 + (-0.516 + 0.856i)T \)
13 \( 1 + (0.198 - 0.980i)T \)
17 \( 1 + (-0.610 + 0.791i)T \)
19 \( 1 + (-0.564 + 0.825i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + (-0.0285 - 0.999i)T \)
31 \( 1 + (-0.870 + 0.491i)T \)
37 \( 1 + (-0.993 - 0.113i)T \)
41 \( 1 + (-0.774 + 0.633i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.254 - 0.967i)T \)
53 \( 1 + (-0.0855 - 0.996i)T \)
59 \( 1 + (-0.774 - 0.633i)T \)
61 \( 1 + (-0.362 + 0.931i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (0.941 - 0.336i)T \)
73 \( 1 + (0.921 + 0.389i)T \)
79 \( 1 + (0.466 - 0.884i)T \)
83 \( 1 + (0.974 + 0.226i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.696 - 0.717i)T \)
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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

−21.90441598798094618238388583303, −20.40281642477326569104950464714, −19.90248994670666882984632519647, −19.06755718077920986266207537508, −18.56229862271534783050701042258, −17.747186717441022770259057347415, −16.81003364772419216785621996635, −16.1826927433930492290144109937, −15.22698533018318492729093167512, −14.01465766313422001892034994143, −13.804429348718108359496639207601, −12.67035132873476966313936349726, −11.97137058801535050028785803785, −11.00689294591364677328921697719, −10.67014636765635054877760428527, −9.27764222831819277201726449107, −8.357748020741571362892304125666, −7.225222291910453158760394824523, −6.94167903670051850503417914880, −6.22159776919261165140514328682, −4.810874224662963242012916222078, −3.87729026861197168923757053419, −2.84214584468277683593616993431, −1.84481860341990637671714647549, −0.44869611263607554686909389747, 0.22699913060039591635200510434, 1.835761000765662401244255597942, 3.30560346215067373953880920580, 3.8362413128396366823447227407, 4.92899490347365123435671723170, 5.664373641396704464858531375340, 6.42557910675171735943662178688, 7.99877571946816919170419116494, 8.52259219587873990908640404962, 9.39143825590870064698670140284, 10.19777455042730884891217888536, 11.08424376349601133820342147498, 11.96741122615239166178635389731, 12.55271791837550219036597778177, 13.42748765818668900962584955589, 14.88409652025987746237778755148, 15.28511447698605912543540837269, 15.96818277898881651570178466978, 16.630402811416925300528122157560, 17.471606102369751802251266421472, 18.3358643565741907591723262042, 19.518572118876867656737381924248, 19.89738151054057067891409367785, 20.906511407962009248433382618329, 21.48370207318740688839623947610

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