Properties

Label 1-731-731.7-r0-0-0
Degree $1$
Conductor $731$
Sign $0.119 - 0.992i$
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  + (0.707 + 0.707i)2-s + (−0.991 − 0.130i)3-s + i·4-s + (−0.130 + 0.991i)5-s + (−0.608 − 0.793i)6-s + (−0.130 − 0.991i)7-s + (−0.707 + 0.707i)8-s + (0.965 + 0.258i)9-s + (−0.793 + 0.608i)10-s + (0.382 − 0.923i)11-s + (0.130 − 0.991i)12-s + (−0.866 + 0.5i)13-s + (0.608 − 0.793i)14-s + (0.258 − 0.965i)15-s − 16-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.991 − 0.130i)3-s + i·4-s + (−0.130 + 0.991i)5-s + (−0.608 − 0.793i)6-s + (−0.130 − 0.991i)7-s + (−0.707 + 0.707i)8-s + (0.965 + 0.258i)9-s + (−0.793 + 0.608i)10-s + (0.382 − 0.923i)11-s + (0.130 − 0.991i)12-s + (−0.866 + 0.5i)13-s + (0.608 − 0.793i)14-s + (0.258 − 0.965i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.119 - 0.992i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.119 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1708849090 - 0.1515376215i\)
\(L(\frac12)\) \(\approx\) \(0.1708849090 - 0.1515376215i\)
\(L(1)\) \(\approx\) \(0.7254034948 + 0.3178299388i\)
\(L(1)\) \(\approx\) \(0.7254034948 + 0.3178299388i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.991 - 0.130i)T \)
5 \( 1 + (-0.130 + 0.991i)T \)
7 \( 1 + (-0.130 - 0.991i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (-0.608 - 0.793i)T \)
29 \( 1 + (0.793 + 0.608i)T \)
31 \( 1 + (-0.991 - 0.130i)T \)
37 \( 1 + (-0.991 - 0.130i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.965 - 0.258i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.130 + 0.991i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.608 - 0.793i)T \)
73 \( 1 + (-0.793 - 0.608i)T \)
79 \( 1 + (-0.991 + 0.130i)T \)
83 \( 1 + (-0.965 + 0.258i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.923 - 0.382i)T \)
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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

−22.64547717135050930803208116547, −21.78639175470755771044347451527, −21.409924598661354696069303011510, −20.363703969372684925625757189833, −19.64680727302053671717708324052, −18.810056026865714583453623907759, −17.71649086737908002638039861734, −17.19313474651113983411517434425, −15.92119823427789328715885496635, −15.42514504436823081329975749481, −14.60074123003455464653431474404, −13.16554954887026249427933909988, −12.58232426897396821924589576267, −12.042642963279223318848965929, −11.46626179283648535677240721432, −10.17158576550037456866399334615, −9.65350386770995477085802360597, −8.66423599231631128439743015393, −7.17601939234467837308327146772, −6.0950391783507609812248714963, −5.314634956685098916878727091, −4.72007186751263522390984304199, −3.85594633923784970053941927674, −2.33353501190408101277992970701, −1.4304015393959030226809293712, 0.09381530014907817882393885911, 2.093908026553054192253461189438, 3.51817560883673747247743940305, 4.17359009064207585849326713422, 5.22459647064538369226454741783, 6.31676963259140396456656011735, 6.79047834366396040149770892888, 7.445000858380615089869639676289, 8.57831455297339280050704974104, 10.12140864466375420062991361752, 10.78431995563289835301955211879, 11.66110921870487009505969400660, 12.38380326331899896827811408514, 13.42369846914787239737088976630, 14.17537500149494383156176140934, 14.825351957628792744916527608688, 15.956116811606269200205609579893, 16.66080976115272339963207283080, 17.15550207189604620032793363712, 18.07199985878645977758033780706, 18.93196889980490654432461712002, 19.853620465971172070873137544699, 21.20812798813726252737662275298, 21.872816345896852477946090145077, 22.432724191609181221713845020142

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