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Properties

Label 1-712-712.221-r1-0-0
Degree $1$
Conductor $712$
Sign $0.0173 - 0.999i$
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  + (0.479 − 0.877i)3-s + (−0.909 − 0.415i)5-s + (−0.349 − 0.936i)7-s + (−0.540 − 0.841i)9-s + (0.415 + 0.909i)11-s + (0.877 + 0.479i)13-s + (−0.800 + 0.599i)15-s + (0.989 − 0.142i)17-s + (0.977 + 0.212i)19-s + (−0.989 − 0.142i)21-s + (0.212 − 0.977i)23-s + (0.654 + 0.755i)25-s + (−0.997 + 0.0713i)27-s + (0.936 − 0.349i)29-s + (0.212 + 0.977i)31-s + ⋯
L(s)  = 1  + (0.479 − 0.877i)3-s + (−0.909 − 0.415i)5-s + (−0.349 − 0.936i)7-s + (−0.540 − 0.841i)9-s + (0.415 + 0.909i)11-s + (0.877 + 0.479i)13-s + (−0.800 + 0.599i)15-s + (0.989 − 0.142i)17-s + (0.977 + 0.212i)19-s + (−0.989 − 0.142i)21-s + (0.212 − 0.977i)23-s + (0.654 + 0.755i)25-s + (−0.997 + 0.0713i)27-s + (0.936 − 0.349i)29-s + (0.212 + 0.977i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0173 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0173 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.0173 - 0.999i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.0173 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.683972623 - 1.655071263i\)
\(L(\frac12)\) \(\approx\) \(1.683972623 - 1.655071263i\)
\(L(1)\) \(\approx\) \(1.115616769 - 0.5266962742i\)
\(L(1)\) \(\approx\) \(1.115616769 - 0.5266962742i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.479 - 0.877i)T \)
5 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (-0.349 - 0.936i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (0.877 + 0.479i)T \)
17 \( 1 + (0.989 - 0.142i)T \)
19 \( 1 + (0.977 + 0.212i)T \)
23 \( 1 + (0.212 - 0.977i)T \)
29 \( 1 + (0.936 - 0.349i)T \)
31 \( 1 + (0.212 + 0.977i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.877 - 0.479i)T \)
43 \( 1 + (0.936 + 0.349i)T \)
47 \( 1 + (0.281 - 0.959i)T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (-0.479 - 0.877i)T \)
61 \( 1 + (0.0713 + 0.997i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.909 + 0.415i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (0.540 - 0.841i)T \)
83 \( 1 + (-0.800 - 0.599i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−22.52628734033375632597507548860, −21.75308271518442412033553799183, −21.09728831556670464544005529178, −20.108286294319729085439685547558, −19.31569687906358017481861985949, −18.83605663979572332922525489667, −17.807323771890353277040026720667, −16.42547005204604385651091722016, −15.95552861222741758334760151114, −15.349568674112386299646035627065, −14.50125865746780764741213070724, −13.744658957831668013391843631108, −12.56950001141340174491113817282, −11.49189163369894507845136451393, −11.07058862008947847130444824886, −9.90861688818731615167183353974, −9.0920235968200993226714228474, −8.28434832547530874606155120882, −7.56869887430029504429484280678, −6.08449370853985011364478614414, −5.42280587964336320836530353440, −4.08191982024030519281816508938, −3.30075966221300356119957432120, −2.758111220143231105844161547237, −0.892122273929827771671765814612, 0.75983838070045639570627469665, 1.32740984619624617751966295821, 2.91142926669373013968861614887, 3.79447804084658248626979529303, 4.62447965320576590427080280437, 6.13686637039567932343590117215, 7.07643911750383029162459564827, 7.608167862285715022939244974027, 8.51511474627406377661354366156, 9.39858921436732269207797410857, 10.45892742068131824212928832985, 11.67367689313483768772715054835, 12.242790302458578061238404174886, 13.02109446365387433973979371372, 13.97843397850977515937319115083, 14.54208896197279757108165445719, 15.71485887913794465186161387211, 16.457178952880884749985789406115, 17.30939022518848042385618671119, 18.29705965724349395314384553976, 19.05189472750597142449085166839, 19.801672568196436015931339905901, 20.393452129005590531285755877086, 20.96852295614007792195778551446, 22.61541906197575059590373728804

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