↑

Properties

Label 1-712-712.413-r0-0-0
Degree $1$
Conductor $712$
Sign $0.976 - 0.213i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 − 0.540i)3-s + (0.654 + 0.755i)5-s + (0.654 + 0.755i)7-s + (0.415 − 0.909i)9-s + (0.654 − 0.755i)11-s + (0.841 − 0.540i)13-s + (0.959 + 0.281i)15-s + (−0.959 + 0.281i)17-s + (0.415 − 0.909i)19-s + (0.959 + 0.281i)21-s + (−0.415 + 0.909i)23-s + (−0.142 + 0.989i)25-s + (−0.142 − 0.989i)27-s + (−0.654 − 0.755i)29-s + (−0.415 − 0.909i)31-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)3-s + (0.654 + 0.755i)5-s + (0.654 + 0.755i)7-s + (0.415 − 0.909i)9-s + (0.654 − 0.755i)11-s + (0.841 − 0.540i)13-s + (0.959 + 0.281i)15-s + (−0.959 + 0.281i)17-s + (0.415 − 0.909i)19-s + (0.959 + 0.281i)21-s + (−0.415 + 0.909i)23-s + (−0.142 + 0.989i)25-s + (−0.142 − 0.989i)27-s + (−0.654 − 0.755i)29-s + (−0.415 − 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.976 - 0.213i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.976 - 0.213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.454250318 - 0.2651504334i\)
\(L(\frac12)\) \(\approx\) \(2.454250318 - 0.2651504334i\)
\(L(1)\) \(\approx\) \(1.706163833 - 0.1185702588i\)
\(L(1)\) \(\approx\) \(1.706163833 - 0.1185702588i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−22.399625550404153184245778072714, −21.69219682775650290134419101630, −20.66393396781927202847605587242, −20.43257922142257096520718518403, −19.79951835251495136873322125420, −18.45517662668470492723960581480, −17.76812339355803127186890792854, −16.593824245551282568783956003649, −16.35936351610134926854518179825, −15.06663203073924191356367639599, −14.290605849675953009336042884417, −13.71709672560772378748158927294, −12.931302735201077430362870826089, −11.803044001892545440822731937046, −10.69204252274671062812444153418, −9.965996128560335749691760999311, −9.03318869474144864149328068884, −8.51652077244394203558280761641, −7.44455440920091552781692189093, −6.425447663586793536290783981234, −5.04896466265706197553173989901, −4.38766204226271814363885184001, −3.60171435641084742087939981726, −2.01198142142934293642390303881, −1.456948805517441333337297508838, 1.31716433083164501928846727291, 2.240240901990983086387976239463, 3.06897609885975266176377137360, 4.07555894671047965781601037688, 5.68622026719853290347486262925, 6.24582659185836815568627059232, 7.3002695527098758118779942918, 8.246280873503951770179273538988, 9.01672604888146167052756321122, 9.72028254545673656946825628459, 11.15903117772517766878001429505, 11.50389841493777582590170996828, 12.94982058642009244654958755977, 13.562190992946117860726771890952, 14.224724884977445816186974221483, 15.16160939953454339778328693508, 15.58347491753336780130276616935, 17.17543243525576558538842209049, 17.92421912435983923824204240759, 18.46529505491694212505076438343, 19.2128224286523326705099832786, 20.12231563711354558664338687477, 20.93854189149306669667966502933, 21.84068999853387585143160396099, 22.24377924167959006090739860553

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