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Properties

Label 1-731-731.117-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.987 - 0.156i$
Analytic cond. $3.39474$
Root an. cond. $3.39474$
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  + (0.781 + 0.623i)2-s + (0.399 + 0.916i)3-s + (0.222 + 0.974i)4-s + (0.757 + 0.652i)5-s + (−0.258 + 0.965i)6-s + (−0.965 + 0.258i)7-s + (−0.433 + 0.900i)8-s + (−0.680 + 0.733i)9-s + (0.185 + 0.982i)10-s + (0.846 − 0.532i)11-s + (−0.804 + 0.593i)12-s + (−0.826 + 0.563i)13-s + (−0.916 − 0.399i)14-s + (−0.294 + 0.955i)15-s + (−0.900 + 0.433i)16-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + (0.399 + 0.916i)3-s + (0.222 + 0.974i)4-s + (0.757 + 0.652i)5-s + (−0.258 + 0.965i)6-s + (−0.965 + 0.258i)7-s + (−0.433 + 0.900i)8-s + (−0.680 + 0.733i)9-s + (0.185 + 0.982i)10-s + (0.846 − 0.532i)11-s + (−0.804 + 0.593i)12-s + (−0.826 + 0.563i)13-s + (−0.916 − 0.399i)14-s + (−0.294 + 0.955i)15-s + (−0.900 + 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.987 - 0.156i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.987 - 0.156i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1791526524 + 2.278914369i\)
\(L(\frac12)\) \(\approx\) \(-0.1791526524 + 2.278914369i\)
\(L(1)\) \(\approx\) \(0.9786062626 + 1.407428828i\)
\(L(1)\) \(\approx\) \(0.9786062626 + 1.407428828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.781 + 0.623i)T \)
3 \( 1 + (0.399 + 0.916i)T \)
5 \( 1 + (0.757 + 0.652i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
11 \( 1 + (0.846 - 0.532i)T \)
13 \( 1 + (-0.826 + 0.563i)T \)
19 \( 1 + (-0.680 - 0.733i)T \)
23 \( 1 + (-0.884 - 0.467i)T \)
29 \( 1 + (0.916 + 0.399i)T \)
31 \( 1 + (0.804 - 0.593i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + (0.993 - 0.111i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (-0.563 + 0.826i)T \)
59 \( 1 + (0.433 + 0.900i)T \)
61 \( 1 + (-0.804 - 0.593i)T \)
67 \( 1 + (-0.733 + 0.680i)T \)
71 \( 1 + (-0.884 + 0.467i)T \)
73 \( 1 + (0.185 - 0.982i)T \)
79 \( 1 + (0.965 - 0.258i)T \)
83 \( 1 + (0.930 + 0.365i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
97 \( 1 + (0.532 + 0.846i)T \)
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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

−22.14722100619647858177240220907, −21.24265463747794673110111609544, −20.348870419447930641237633215784, −19.65861242885852072749077628288, −19.36979785416040168378186704313, −18.10379999408649781145715011300, −17.376109533584026363812977516163, −16.4164013447445714237769472389, −15.24329391668096924061263528401, −14.30404986039401531907400260870, −13.77209311584810930706721148438, −12.82128312688130837121800916941, −12.4853743595946889436555611387, −11.75310635623762855124181590190, −10.17349184056816526927507004023, −9.7471755810422523060874148338, −8.79439171610248567982438033842, −7.50917063998099632734669592201, −6.36078585125519967515200508438, −6.03459563966801001004322806707, −4.720829759767446805125790813225, −3.68766866310974483027331975352, −2.60713364622956588955339712337, −1.82354159174452932822384028745, −0.74575583453841394134372548849, 2.4179032768700272753310935846, 2.88538597289123682142684493147, 3.98706907996687016438700755103, 4.763092026577509796504415779307, 6.116243427860687179579786875760, 6.337567950358282721538382898952, 7.55980281046427487377763876947, 8.840313099331683622370016420065, 9.39931717503422967896600641249, 10.35814087990643303907734382808, 11.36777054441273210942999242816, 12.33497210407301883192085187180, 13.43487134241647288767431947609, 14.07599733899466534904452253, 14.69719354201664551099964080615, 15.45260644914076877914821859146, 16.36167352540057220761932337458, 16.9040468268962411646107916783, 17.76922875604679795356736217977, 19.13544617304112287529676620802, 19.74204089630500139843275256828, 20.89304105332394839858518721391, 21.72104477268664102899006611033, 22.09371929190673867975548525929, 22.5436052457606867155894470409

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