Properties

Label 1-712-712.579-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.735 + 0.677i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (0.415 − 0.909i)3-s + (0.142 − 0.989i)5-s + (0.142 − 0.989i)7-s + (−0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.415 + 0.909i)13-s + (−0.841 − 0.540i)15-s + (0.841 − 0.540i)17-s + (−0.654 − 0.755i)19-s + (−0.841 − 0.540i)21-s + (0.654 + 0.755i)23-s + (−0.959 − 0.281i)25-s + (−0.959 + 0.281i)27-s + (0.142 − 0.989i)29-s + (0.654 − 0.755i)31-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)3-s + (0.142 − 0.989i)5-s + (0.142 − 0.989i)7-s + (−0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.415 + 0.909i)13-s + (−0.841 − 0.540i)15-s + (0.841 − 0.540i)17-s + (−0.654 − 0.755i)19-s + (−0.841 − 0.540i)21-s + (0.654 + 0.755i)23-s + (−0.959 − 0.281i)25-s + (−0.959 + 0.281i)27-s + (0.142 − 0.989i)29-s + (0.654 − 0.755i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.735 + 0.677i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.735 + 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6491988095 - 1.662417565i\)
\(L(\frac12)\) \(\approx\) \(-0.6491988095 - 1.662417565i\)
\(L(1)\) \(\approx\) \(0.8039737765 - 0.8361792549i\)
\(L(1)\) \(\approx\) \(0.8039737765 - 0.8361792549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.415 - 0.909i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (0.142 - 0.989i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 - T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.415 - 0.909i)T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.654 + 0.755i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−22.54069437087653809702737257777, −22.24558294424043792232494061526, −21.1111670769970777919829043159, −20.83214814939621397462475995644, −19.51873666071992892178963650637, −19.0061706244837288408571769664, −17.94969442661474519938129270002, −17.31013270160071962277565693440, −16.11101596773182672023445462436, −15.3287234838474939967046742773, −14.62694322411760971391266769286, −14.413646789843078340318035208639, −12.83727929695208195542738298757, −12.16225340213057881369753541030, −10.90827880000840966268182284877, −10.29386101905539777144684427033, −9.65809256505920148703300711304, −8.53639382409222768082522430162, −7.815786763943127402120238427390, −6.629162818480254342558727628049, −5.55050009756183294718801449298, −4.815260208989143426317080272191, −3.51367621128587109098075312894, −2.784110153343169722221494295752, −1.90129234351190665312483827739, 0.40516594598534980250706593618, 1.08886558660410433083327041725, 2.208473229834739340039095721013, 3.435889926381460933537083683035, 4.4837846590330084544579499724, 5.5431132381704080420149744376, 6.60841761197721033870397361633, 7.47514806265031731087548057298, 8.27197035561694996444303336357, 9.07101546458740332580129463865, 9.938918318893035716620552764692, 11.31998786632447805754086177304, 11.90391402154791190067356080870, 12.99744070945524523947732447121, 13.62812922196012710842416780291, 14.05920964748187438801295706447, 15.25424835426171473958045126978, 16.44648432408050109621040665186, 17.03863952051188257626279229918, 17.64610235425738269691249289775, 19.02209330870497230955823768767, 19.2658639087298277849097951966, 20.27161069700834380027707776643, 20.951806044596987552299731102424, 21.62299385919050688409467898176

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