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Properties

Label 1-712-712.53-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.557 - 0.829i$
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  + (−0.755 + 0.654i)3-s + (−0.959 − 0.281i)5-s + (0.281 − 0.959i)7-s + (0.142 − 0.989i)9-s + (0.959 − 0.281i)11-s + (0.755 − 0.654i)13-s + (0.909 − 0.415i)15-s + (−0.415 + 0.909i)17-s + (−0.989 − 0.142i)19-s + (0.415 + 0.909i)21-s + (−0.989 − 0.142i)23-s + (0.841 + 0.540i)25-s + (0.540 + 0.841i)27-s + (0.281 − 0.959i)29-s + (−0.989 + 0.142i)31-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)3-s + (−0.959 − 0.281i)5-s + (0.281 − 0.959i)7-s + (0.142 − 0.989i)9-s + (0.959 − 0.281i)11-s + (0.755 − 0.654i)13-s + (0.909 − 0.415i)15-s + (−0.415 + 0.909i)17-s + (−0.989 − 0.142i)19-s + (0.415 + 0.909i)21-s + (−0.989 − 0.142i)23-s + (0.841 + 0.540i)25-s + (0.540 + 0.841i)27-s + (0.281 − 0.959i)29-s + (−0.989 + 0.142i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.557 - 0.829i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.557 - 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2322031289 - 0.4358891068i\)
\(L(\frac12)\) \(\approx\) \(0.2322031289 - 0.4358891068i\)
\(L(1)\) \(\approx\) \(0.6536049073 - 0.08932254744i\)
\(L(1)\) \(\approx\) \(0.6536049073 - 0.08932254744i\)

Euler product

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

−22.92736698453606971820830495760, −22.16812762504421672789966567795, −21.56626225541448403847330289486, −20.22581383644400909345478714415, −19.50789797891435847778405523321, −18.55109971376665209553329154073, −18.312794548979355570331013367431, −17.22042772798390227431369672738, −16.273007664497034970760767389865, −15.67595001613931964192433337378, −14.62976680695247490131182245381, −13.85926034088656691043233569036, −12.543879217802230274957252387477, −12.08547418808343642440879457972, −11.33618735929452860059779160085, −10.75878768497481511679911691720, −9.21187164236848200696035056187, −8.47715322994295467423892550446, −7.43624325642219990831876489357, −6.65337582077048553875051399349, −5.88749365679268999252591057232, −4.71426137294907190485085805607, −3.86099039018900173463120969040, −2.40930909513434789801561993234, −1.423858270193795467628746933323, 0.28459701666749751088081720858, 1.46489614927435840725261359862, 3.58347009561130926793775274416, 3.98018782734999064992219082605, 4.78135911324044903092970616403, 6.07307543289516950721843248724, 6.75942394010060077134106472178, 8.03487383199061749954101798654, 8.68548806704026594110836819150, 9.915620072096396848430664723279, 10.835985683296597153591547323124, 11.2539488629960371344131651538, 12.21009442443368647505201362060, 13.046820370294337608838651220, 14.216322970398360572191756159331, 15.162564248092430118369536924117, 15.751346232431729220827324131944, 16.78091280861271976239985332798, 17.09822374882578943731420791283, 18.0743491949498658575741918044, 19.219692578950161517067895014317, 20.05143011853937661325633817514, 20.58460615494854263432685525562, 21.609786896437082033571074147887, 22.39330458119808407155747051661

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