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Properties

Label 1-1712-1712.1333-r0-0-0
Degree $1$
Conductor $1712$
Sign $0.834 + 0.551i$
Analytic cond. $7.95049$
Root an. cond. $7.95049$
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  + (−0.964 − 0.263i)3-s + (−0.737 + 0.674i)5-s + (−0.757 + 0.652i)7-s + (0.861 + 0.508i)9-s + (0.812 + 0.582i)11-s + (0.776 + 0.630i)13-s + (0.889 − 0.456i)15-s + (−0.984 − 0.176i)17-s + (0.978 + 0.205i)19-s + (0.902 − 0.430i)21-s + (0.320 − 0.947i)23-s + (0.0887 − 0.996i)25-s + (−0.696 − 0.717i)27-s + (−0.234 − 0.972i)29-s + (0.829 − 0.558i)31-s + ⋯
L(s)  = 1  + (−0.964 − 0.263i)3-s + (−0.737 + 0.674i)5-s + (−0.757 + 0.652i)7-s + (0.861 + 0.508i)9-s + (0.812 + 0.582i)11-s + (0.776 + 0.630i)13-s + (0.889 − 0.456i)15-s + (−0.984 − 0.176i)17-s + (0.978 + 0.205i)19-s + (0.902 − 0.430i)21-s + (0.320 − 0.947i)23-s + (0.0887 − 0.996i)25-s + (−0.696 − 0.717i)27-s + (−0.234 − 0.972i)29-s + (0.829 − 0.558i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1712\)    =    \(2^{4} \cdot 107\)
Sign: $0.834 + 0.551i$
Analytic conductor: \(7.95049\)
Root analytic conductor: \(7.95049\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1712} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1712,\ (0:\ ),\ 0.834 + 0.551i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8937486971 + 0.2684504767i\)
\(L(\frac12)\) \(\approx\) \(0.8937486971 + 0.2684504767i\)
\(L(1)\) \(\approx\) \(0.7283533781 + 0.1121505843i\)
\(L(1)\) \(\approx\) \(0.7283533781 + 0.1121505843i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
107 \( 1 \)
good3 \( 1 + (-0.964 - 0.263i)T \)
5 \( 1 + (-0.737 + 0.674i)T \)
7 \( 1 + (-0.757 + 0.652i)T \)
11 \( 1 + (0.812 + 0.582i)T \)
13 \( 1 + (0.776 + 0.630i)T \)
17 \( 1 + (-0.984 - 0.176i)T \)
19 \( 1 + (0.978 + 0.205i)T \)
23 \( 1 + (0.320 - 0.947i)T \)
29 \( 1 + (-0.234 - 0.972i)T \)
31 \( 1 + (0.829 - 0.558i)T \)
37 \( 1 + (0.875 + 0.482i)T \)
41 \( 1 + (0.956 - 0.292i)T \)
43 \( 1 + (0.737 + 0.674i)T \)
47 \( 1 + (-0.956 - 0.292i)T \)
53 \( 1 + (-0.926 + 0.375i)T \)
59 \( 1 + (-0.234 + 0.972i)T \)
61 \( 1 + (0.652 - 0.757i)T \)
67 \( 1 + (-0.403 + 0.915i)T \)
71 \( 1 + (-0.263 - 0.964i)T \)
73 \( 1 + (0.533 - 0.845i)T \)
79 \( 1 + (-0.998 - 0.0592i)T \)
83 \( 1 + (0.989 - 0.147i)T \)
89 \( 1 + (0.794 - 0.606i)T \)
97 \( 1 + (0.937 - 0.348i)T \)
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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

−20.109935643075286147732497576, −19.65162185525614327602125015575, −18.82701531273383722298415748012, −17.7579747021451103111377279533, −17.293275796580518743704228774687, −16.354332735486158788486637527374, −15.985151389515928475458594009012, −15.47159063267041456808066156744, −14.25005188633289875740642745881, −13.17543921482582825039287479671, −12.85604644256442157832313410258, −11.8323440419877859687863330464, −11.21236311642574203349812530019, −10.67615843801518119375039777401, −9.52561548191346110458840433201, −9.038528019991471601750957889606, −7.92484771075451186317336416219, −7.047894340074902078727769431186, −6.32459986222196165260492339083, −5.50001563163127202189879215136, −4.60327771238882123342781700724, −3.78656630067221555425730626389, −3.25574375330793263331964708115, −1.26396311099750046888541639353, −0.716943459742122696676387005451, 0.70109204361098659650416775978, 2.00009436926947476853944194395, 2.96993739646552362338623455429, 4.11544309570771392373961226880, 4.62485197615190675153175777763, 6.09841293655177740898433959780, 6.346746665793452329999794034581, 7.11494426180256283560003240514, 7.97795588153347554455764970742, 9.11808190676926717453291443046, 9.79951897232565760936286037127, 10.77246307651804680059035217704, 11.601088196116685762473383319604, 11.82653420581521929084107285893, 12.76781940457473098461323966185, 13.512212670491518792754692243028, 14.53615885891164618606268874633, 15.39174437379380853593748201573, 15.97950055337735037607831005062, 16.54597842856801811071711589664, 17.55700315143242656671181304440, 18.23580656591838278997389198165, 18.85787273673315671830895106810, 19.37680552797189799823805136727, 20.26826114775540289243190411325

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