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Properties

Label 1-712-712.261-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.980 - 0.196i$
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  + (0.349 + 0.936i)3-s + (0.989 + 0.142i)5-s + (0.599 + 0.800i)7-s + (−0.755 + 0.654i)9-s + (−0.142 − 0.989i)11-s + (−0.936 + 0.349i)13-s + (0.212 + 0.977i)15-s + (0.540 + 0.841i)17-s + (0.0713 − 0.997i)19-s + (−0.540 + 0.841i)21-s + (−0.997 − 0.0713i)23-s + (0.959 + 0.281i)25-s + (−0.877 − 0.479i)27-s + (−0.800 + 0.599i)29-s + (−0.997 + 0.0713i)31-s + ⋯
L(s)  = 1  + (0.349 + 0.936i)3-s + (0.989 + 0.142i)5-s + (0.599 + 0.800i)7-s + (−0.755 + 0.654i)9-s + (−0.142 − 0.989i)11-s + (−0.936 + 0.349i)13-s + (0.212 + 0.977i)15-s + (0.540 + 0.841i)17-s + (0.0713 − 0.997i)19-s + (−0.540 + 0.841i)21-s + (−0.997 − 0.0713i)23-s + (0.959 + 0.281i)25-s + (−0.877 − 0.479i)27-s + (−0.800 + 0.599i)29-s + (−0.997 + 0.0713i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.980 - 0.196i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.980 - 0.196i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1414022226 + 1.423688138i\)
\(L(\frac12)\) \(\approx\) \(-0.1414022226 + 1.423688138i\)
\(L(1)\) \(\approx\) \(1.038352899 + 0.5956886581i\)
\(L(1)\) \(\approx\) \(1.038352899 + 0.5956886581i\)

Euler product

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

−21.99653269067752672062837260957, −20.84192210217042727460586139001, −20.36065857531521542398420950376, −19.73497640530818267533032402784, −18.297725122633053841109975822579, −18.14286732672235843952530836947, −17.13217831066583201997263423872, −16.63980191644237055474847520647, −15.00801595627280915930284687347, −14.3475773407828112840803312486, −13.793093126470804521664631614712, −12.83062577866757199867476348227, −12.251678939380267164559329125402, −11.17167568475775682246773164474, −9.86833356156341212714725521934, −9.594802704792017338939760745920, −8.05630837745023603910371055824, −7.586868612445119409390288545436, −6.68706678611406110346937866162, −5.61699793093078499655866643414, −4.75702914712422348593516233378, −3.38531983251230536119131430784, −2.09650046610032728709620703407, −1.61968232525335344663442209634, −0.26679643629143134820552147732, 1.721406521550314552660696642996, 2.57448954017032396268024822820, 3.522818021851036051818973226135, 4.879504334905788540603143841301, 5.443238643706456112701845009823, 6.32107234550892660031291952885, 7.78295495888329397778092150973, 8.7152542710177305833227920183, 9.32734626272290868666590700707, 10.20195783201716933365103229064, 10.99589815949907113309776579560, 11.85781434329376850795012951456, 13.06877403207686805464088126633, 13.97951533171359209292920205919, 14.65683270313651909257793014076, 15.22949974403515650180928274013, 16.41320790603063105680876610270, 16.94792237337227404345621687136, 17.91963108493624359287587346517, 18.74501106240744555269552132727, 19.670943456713958097585468242841, 20.59638033947922266771189099668, 21.4583432164918613997120832353, 21.89042800295145689937261635424, 22.22272172769179402009564864491

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