↑

Properties

Label 1-731-731.566-r0-0-0
Degree $1$
Conductor $731$
Sign $0.838 + 0.544i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (0.608 + 0.793i)3-s − i·4-s + (−0.793 + 0.608i)5-s + (0.991 + 0.130i)6-s + (−0.793 − 0.608i)7-s + (−0.707 − 0.707i)8-s + (−0.258 + 0.965i)9-s + (−0.130 + 0.991i)10-s + (0.382 + 0.923i)11-s + (0.793 − 0.608i)12-s + (0.866 − 0.5i)13-s + (−0.991 + 0.130i)14-s + (−0.965 − 0.258i)15-s − 16-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.608 + 0.793i)3-s − i·4-s + (−0.793 + 0.608i)5-s + (0.991 + 0.130i)6-s + (−0.793 − 0.608i)7-s + (−0.707 − 0.707i)8-s + (−0.258 + 0.965i)9-s + (−0.130 + 0.991i)10-s + (0.382 + 0.923i)11-s + (0.793 − 0.608i)12-s + (0.866 − 0.5i)13-s + (−0.991 + 0.130i)14-s + (−0.965 − 0.258i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.838 + 0.544i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.838 + 0.544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.860997942 + 0.5514870924i\)
\(L(\frac12)\) \(\approx\) \(1.860997942 + 0.5514870924i\)
\(L(1)\) \(\approx\) \(1.495244246 + 0.003955047396i\)
\(L(1)\) \(\approx\) \(1.495244246 + 0.003955047396i\)

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 \( 1 + (0.608 + 0.793i)T \)
5 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 + (-0.793 - 0.608i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (0.991 + 0.130i)T \)
29 \( 1 + (0.130 + 0.991i)T \)
31 \( 1 + (0.608 + 0.793i)T \)
37 \( 1 + (0.608 + 0.793i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.258 - 0.965i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (0.793 + 0.608i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.991 + 0.130i)T \)
73 \( 1 + (-0.130 - 0.991i)T \)
79 \( 1 + (0.608 - 0.793i)T \)
83 \( 1 + (0.258 + 0.965i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.923 + 0.382i)T \)
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

−22.71523025431522381032119368104, −21.69982852575487020378106648281, −20.88162283930719109594817509660, −20.0354359453807286092922586190, −19.096312170895969773114595465305, −18.65293743814142743465631620044, −17.36946082024071115989458533130, −16.55293492855164090869612055905, −15.70226487737989997522630990150, −15.22227917129015548261656145979, −14.11543873265316110348688296491, −13.25615547813078982104257906211, −12.92583310575804889397804315632, −11.77198179562599824443356781823, −11.44713332787391171155275081168, −9.26269706914462420440575011159, −8.75448937307715125745804555115, −8.08650280700327798432460257813, −7.00787926825146895209301418145, −6.35449588898597383885890482006, −5.43904050258019958856283028404, −4.09062948834629779602653508210, −3.39014176016182199104848767584, −2.47512487468293635631246311125, −0.73352143304935365556931971696, 1.35465846057993823471566873207, 2.901979052596368490061097485, 3.40005039978193523284126676957, 4.12363651890072805409597213084, 5.00458979947080482357103217658, 6.347086742849377995645527372398, 7.21270844853562572076747894110, 8.39575587784880760701122910393, 9.51834258123468111227781856635, 10.27027670067651432896185470541, 10.788231316628453366866152005955, 11.77064297412707363731281644938, 12.751854678518408094936456973311, 13.56776812018847259542803963637, 14.46947090411181352481217238485, 15.04464397598964951919254518169, 15.809922518772913942419914545776, 16.50811725605494530994644733850, 17.98144488448133242187799952307, 19.02419611203968402879234594764, 19.54906877148991091286820970361, 20.34812002606819203996347360303, 20.73003703414966832701746767279, 21.894492148549651970213586868491, 22.69375160936463516774546932298

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