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Properties

Label 1-712-712.243-r0-0-0
Degree $1$
Conductor $712$
Sign $0.891 + 0.453i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (0.936 + 0.349i)3-s + (−0.989 + 0.142i)5-s + (0.800 + 0.599i)7-s + (0.755 + 0.654i)9-s + (0.142 − 0.989i)11-s + (0.349 − 0.936i)13-s + (−0.977 − 0.212i)15-s + (−0.540 + 0.841i)17-s + (0.997 − 0.0713i)19-s + (0.540 + 0.841i)21-s + (−0.0713 − 0.997i)23-s + (0.959 − 0.281i)25-s + (0.479 + 0.877i)27-s + (0.599 − 0.800i)29-s + (−0.0713 + 0.997i)31-s + ⋯
L(s)  = 1  + (0.936 + 0.349i)3-s + (−0.989 + 0.142i)5-s + (0.800 + 0.599i)7-s + (0.755 + 0.654i)9-s + (0.142 − 0.989i)11-s + (0.349 − 0.936i)13-s + (−0.977 − 0.212i)15-s + (−0.540 + 0.841i)17-s + (0.997 − 0.0713i)19-s + (0.540 + 0.841i)21-s + (−0.0713 − 0.997i)23-s + (0.959 − 0.281i)25-s + (0.479 + 0.877i)27-s + (0.599 − 0.800i)29-s + (−0.0713 + 0.997i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.891 + 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.900613150 + 0.4554461408i\)
\(L(\frac12)\) \(\approx\) \(1.900613150 + 0.4554461408i\)
\(L(1)\) \(\approx\) \(1.404289721 + 0.2098036737i\)
\(L(1)\) \(\approx\) \(1.404289721 + 0.2098036737i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.936 + 0.349i)T \)
5 \( 1 + (-0.989 + 0.142i)T \)
7 \( 1 + (0.800 + 0.599i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
13 \( 1 + (0.349 - 0.936i)T \)
17 \( 1 + (-0.540 + 0.841i)T \)
19 \( 1 + (0.997 - 0.0713i)T \)
23 \( 1 + (-0.0713 - 0.997i)T \)
29 \( 1 + (0.599 - 0.800i)T \)
31 \( 1 + (-0.0713 + 0.997i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.349 + 0.936i)T \)
43 \( 1 + (-0.599 - 0.800i)T \)
47 \( 1 + (-0.909 - 0.415i)T \)
53 \( 1 + (0.909 - 0.415i)T \)
59 \( 1 + (-0.936 + 0.349i)T \)
61 \( 1 + (-0.877 + 0.479i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (0.989 + 0.142i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (0.755 - 0.654i)T \)
83 \( 1 + (-0.977 + 0.212i)T \)
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

−22.79672224651917147817356709710, −21.47394240233923575219646599752, −20.67085330473361598452096095176, −20.00847092198799873453219218380, −19.61337577561271108071658932366, −18.38918692731020366927782821510, −17.9773137426747930268878914237, −16.73383616466900217372943118033, −15.78936400308837238884982957552, −15.11715341331439958440744092323, −14.21336301857075249785209272016, −13.670416577011423399659066405192, −12.56966030591009880956242838757, −11.73008675790598507710169181733, −11.05523010673883609527243168937, −9.64941405345573583034216258352, −9.03817298195526152514833371220, −7.89015794543403529883512650543, −7.45410778560312795667002200865, −6.703985027941447377675810483145, −4.89895121181694232917674048690, −4.21646976909844185767124957832, −3.37181108401930645113194314124, −2.0627138184206379495451443256, −1.10122221227122983949768196207, 1.16245462213583664712502291834, 2.6152522038561284123361611736, 3.345331717198119082654186981276, 4.29606917798728525435736626800, 5.22230725936674464059697470908, 6.48045785318867731429864117024, 7.77190006051461803763978312572, 8.3440479926255984453277097110, 8.77396234185597944641862921709, 10.157937320375884262030164820892, 10.9652839375176517651293174334, 11.74113885691523426067388476411, 12.76970109041160389385930489152, 13.731895051094371540537407211, 14.589903785000690144389797065698, 15.26812444880649188026679138953, 15.82198955462674285549870768447, 16.71918477567663565783620757143, 18.11908438363123994081579756218, 18.600294024129045604811976156661, 19.65121228756761231455043600027, 20.05376227267220789146092085497, 21.02781152833139660434548806342, 21.738841307100719524271751239314, 22.50888270701943354416426053771

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