Properties

Label 1-731-731.100-r0-0-0
Degree $1$
Conductor $731$
Sign $0.522 - 0.852i$
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 + (0.593 − 0.804i)3-s + (0.222 + 0.974i)4-s + (0.185 − 0.982i)5-s + (0.965 − 0.258i)6-s + (0.258 − 0.965i)7-s + (−0.433 + 0.900i)8-s + (−0.294 − 0.955i)9-s + (0.757 − 0.652i)10-s + (0.846 − 0.532i)11-s + (0.916 + 0.399i)12-s + (−0.0747 − 0.997i)13-s + (0.804 − 0.593i)14-s + (−0.680 − 0.733i)15-s + (−0.900 + 0.433i)16-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + (0.593 − 0.804i)3-s + (0.222 + 0.974i)4-s + (0.185 − 0.982i)5-s + (0.965 − 0.258i)6-s + (0.258 − 0.965i)7-s + (−0.433 + 0.900i)8-s + (−0.294 − 0.955i)9-s + (0.757 − 0.652i)10-s + (0.846 − 0.532i)11-s + (0.916 + 0.399i)12-s + (−0.0747 − 0.997i)13-s + (0.804 − 0.593i)14-s + (−0.680 − 0.733i)15-s + (−0.900 + 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.522 - 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.438068185 - 1.366040338i\)
\(L(\frac12)\) \(\approx\) \(2.438068185 - 1.366040338i\)
\(L(1)\) \(\approx\) \(1.929686170 - 0.3932611274i\)
\(L(1)\) \(\approx\) \(1.929686170 - 0.3932611274i\)

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 \( 1 + (0.593 - 0.804i)T \)
5 \( 1 + (0.185 - 0.982i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
11 \( 1 + (0.846 - 0.532i)T \)
13 \( 1 + (-0.0747 - 0.997i)T \)
19 \( 1 + (-0.294 + 0.955i)T \)
23 \( 1 + (0.0373 + 0.999i)T \)
29 \( 1 + (-0.804 + 0.593i)T \)
31 \( 1 + (-0.916 - 0.399i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (0.993 - 0.111i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.997 + 0.0747i)T \)
59 \( 1 + (0.433 + 0.900i)T \)
61 \( 1 + (0.916 - 0.399i)T \)
67 \( 1 + (0.955 + 0.294i)T \)
71 \( 1 + (0.0373 - 0.999i)T \)
73 \( 1 + (0.757 + 0.652i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 + (-0.149 - 0.988i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
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

−22.19216673280365555940563098624, −21.96353436780278898449454624816, −21.22776826820962203577749652906, −20.41492635694650168204467846426, −19.46569801917478813624049493724, −18.93928232644392563257303938770, −18.07639717459151223404542984290, −16.80438044049926734248566858071, −15.66532901084482355613704164524, −14.96483862262010272372163764048, −14.54317593237056187435411330861, −13.85817303748913247151313555517, −12.7813310269242605733565934860, −11.61338143640300111158962764534, −11.20895980723479246027988673733, −10.16228549399871685775896596816, −9.40779127810603238580666779237, −8.73175722389806955587544128157, −7.13037185526004960303671478850, −6.31421499876434296167041303422, −5.22148205203800627550244504882, −4.32415930050273550555654055491, −3.51502287916413900749343828565, −2.38919174510989104034150708244, −2.01374093187358533356842025373, 0.95345633367200413142870992014, 2.03539153630371245986009412667, 3.552545398196351628650759569675, 3.98298540651753677188194965379, 5.40034558199767980576849832487, 6.03714349846596046216866211426, 7.24782838457276163526008509211, 7.80223919706963153165639263569, 8.6220937168007454114817773391, 9.47541843475082774430199242389, 11.00245785161462660756865002249, 12.00008562379079248017551899307, 12.84022628354296037897725364486, 13.270387269760163172991665326539, 14.19666609710652899743234503087, 14.64881366650634451697756380036, 15.842522435486529585238061845805, 16.761863860840439562850202541301, 17.30689113542182594208619108439, 18.058505055982640752694199173734, 19.45564414314111584631627510728, 20.13210155688744598631462324105, 20.707991230935055289903630250871, 21.49818556318953230469287348162, 22.68739796053843426824392622997

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