Properties

Label 1-968-968.123-r0-0-0
Degree $1$
Conductor $968$
Sign $-0.326 + 0.945i$
Analytic cond. $4.49537$
Root an. cond. $4.49537$
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 + (0.466 + 0.884i)5-s + (−0.921 − 0.389i)7-s + (−0.809 − 0.587i)9-s + (−0.870 + 0.491i)13-s + (0.985 − 0.170i)15-s + (−0.941 + 0.336i)17-s + (0.0285 − 0.999i)19-s + (−0.654 + 0.755i)21-s + (0.654 + 0.755i)23-s + (−0.564 + 0.825i)25-s + (−0.809 + 0.587i)27-s + (−0.564 − 0.825i)29-s + (−0.198 + 0.980i)31-s + (−0.0855 − 0.996i)35-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + (0.466 + 0.884i)5-s + (−0.921 − 0.389i)7-s + (−0.809 − 0.587i)9-s + (−0.870 + 0.491i)13-s + (0.985 − 0.170i)15-s + (−0.941 + 0.336i)17-s + (0.0285 − 0.999i)19-s + (−0.654 + 0.755i)21-s + (0.654 + 0.755i)23-s + (−0.564 + 0.825i)25-s + (−0.809 + 0.587i)27-s + (−0.564 − 0.825i)29-s + (−0.198 + 0.980i)31-s + (−0.0855 − 0.996i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.326 + 0.945i$
Analytic conductor: \(4.49537\)
Root analytic conductor: \(4.49537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 968,\ (0:\ ),\ -0.326 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2825846061 + 0.3964713442i\)
\(L(\frac12)\) \(\approx\) \(0.2825846061 + 0.3964713442i\)
\(L(1)\) \(\approx\) \(0.8432262843 - 0.06815273183i\)
\(L(1)\) \(\approx\) \(0.8432262843 - 0.06815273183i\)

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 \( 1 + (0.466 + 0.884i)T \)
7 \( 1 + (-0.921 - 0.389i)T \)
13 \( 1 + (-0.870 + 0.491i)T \)
17 \( 1 + (-0.941 + 0.336i)T \)
19 \( 1 + (0.0285 - 0.999i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (-0.198 + 0.980i)T \)
37 \( 1 + (0.736 - 0.676i)T \)
41 \( 1 + (0.254 + 0.967i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (-0.774 + 0.633i)T \)
53 \( 1 + (-0.974 + 0.226i)T \)
59 \( 1 + (-0.254 + 0.967i)T \)
61 \( 1 + (-0.998 + 0.0570i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.610 + 0.791i)T \)
73 \( 1 + (-0.516 + 0.856i)T \)
79 \( 1 + (0.696 - 0.717i)T \)
83 \( 1 + (-0.0855 + 0.996i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.466 + 0.884i)T \)
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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

−21.55177167820988138518558733876, −20.51722549406112091161422508864, −20.26634271463737746982262348654, −19.34126418370943392205498894909, −18.430624046464585532444229014971, −17.2421309198650295340229762480, −16.70377913232191924039061181249, −16.03587120078423550018338348002, −15.24907370993006706112062277733, −14.497212829169292032194343896237, −13.46799893719090658943643104433, −12.78369244327159058280970597674, −12.00741744985028511127235131812, −10.81930984547711791107566279373, −9.971005855480810359968961949685, −9.360993997357551929936273370772, −8.76867758020307833321948944011, −7.8056124187787473718986329586, −6.49164274819409467881196446877, −5.536979700031632964683083332691, −4.8882126882019886146085629006, −3.92043188122991669712613136717, −2.884484597593381204902304888330, −2.0231948474681666182258905307, −0.18488443486283536058881544446, 1.46046782851532185272644183260, 2.579843629540949588201553827191, 3.067075019293753581446315760570, 4.33973135429827875642910984891, 5.74669188287498695342774498726, 6.618920790092926409036661471065, 7.02037517400215035042098439964, 7.85240234906334539147117177277, 9.26339370952115476201914099368, 9.55954791335821994832721909017, 10.87503926569577613864084648226, 11.46682436345686449541736247392, 12.68038665228566017066948564487, 13.228810212715484019161544632883, 13.90540346017460964392990735857, 14.72657380508735658428180116964, 15.45958406444575518761367919090, 16.65991549467763568445254088632, 17.520180226170499139256604343341, 17.971585375196806021807012433889, 19.066718678767831332843721676533, 19.47023598278109305481959633864, 20.05656669778745247425809663045, 21.37026308188444311906151440632, 22.00410453071278419719870990709

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