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Properties

Label 1-731-731.158-r0-0-0
Degree $1$
Conductor $731$
Sign $0.953 + 0.302i$
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  + (−0.111 − 0.993i)2-s + (−0.240 + 0.970i)3-s + (−0.974 + 0.222i)4-s + (0.978 − 0.204i)5-s + (0.991 + 0.130i)6-s + (−0.793 − 0.608i)7-s + (0.330 + 0.943i)8-s + (−0.884 − 0.467i)9-s + (−0.312 − 0.949i)10-s + (−0.985 + 0.167i)11-s + (0.0186 − 0.999i)12-s + (−0.997 + 0.0747i)13-s + (−0.516 + 0.856i)14-s + (−0.0373 + 0.999i)15-s + (0.900 − 0.433i)16-s + ⋯
L(s)  = 1  + (−0.111 − 0.993i)2-s + (−0.240 + 0.970i)3-s + (−0.974 + 0.222i)4-s + (0.978 − 0.204i)5-s + (0.991 + 0.130i)6-s + (−0.793 − 0.608i)7-s + (0.330 + 0.943i)8-s + (−0.884 − 0.467i)9-s + (−0.312 − 0.949i)10-s + (−0.985 + 0.167i)11-s + (0.0186 − 0.999i)12-s + (−0.997 + 0.0747i)13-s + (−0.516 + 0.856i)14-s + (−0.0373 + 0.999i)15-s + (0.900 − 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.953 + 0.302i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.953 + 0.302i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8898614947 + 0.1379672201i\)
\(L(\frac12)\) \(\approx\) \(0.8898614947 + 0.1379672201i\)
\(L(1)\) \(\approx\) \(0.8067222505 - 0.1140707803i\)
\(L(1)\) \(\approx\) \(0.8067222505 - 0.1140707803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.111 - 0.993i)T \)
3 \( 1 + (-0.240 + 0.970i)T \)
5 \( 1 + (0.978 - 0.204i)T \)
7 \( 1 + (-0.793 - 0.608i)T \)
11 \( 1 + (-0.985 + 0.167i)T \)
13 \( 1 + (-0.997 + 0.0747i)T \)
19 \( 1 + (0.884 - 0.467i)T \)
23 \( 1 + (-0.347 + 0.937i)T \)
29 \( 1 + (0.856 + 0.516i)T \)
31 \( 1 + (0.999 + 0.0186i)T \)
37 \( 1 + (0.608 + 0.793i)T \)
41 \( 1 + (-0.276 + 0.960i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (0.652 + 0.757i)T \)
59 \( 1 + (-0.943 - 0.330i)T \)
61 \( 1 + (0.0186 + 0.999i)T \)
67 \( 1 + (-0.955 - 0.294i)T \)
71 \( 1 + (0.347 + 0.937i)T \)
73 \( 1 + (-0.312 + 0.949i)T \)
79 \( 1 + (0.608 - 0.793i)T \)
83 \( 1 + (-0.593 + 0.804i)T \)
89 \( 1 + (-0.149 - 0.988i)T \)
97 \( 1 + (-0.578 + 0.815i)T \)
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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

−22.52113657810071820066858302992, −22.135955183808741446955483438266, −21.046611092413933191323817288007, −19.69215009382594065817548122368, −18.86148301881639326632946247638, −18.296810476234697829691054405184, −17.68686422432634899690342365085, −16.84410478456608575736196997824, −16.10492412304955307517787615497, −15.12095718464907342701299929000, −14.09072317093865978855371228752, −13.606318233881569231582948670503, −12.695135726649166313261999153825, −12.15112895856634001038951845529, −10.491095852304655122473176301679, −9.83051170935453940519801450188, −8.865280525120318222923694995897, −7.916693760553550280789860942218, −7.09268045512556734520303235536, −6.21061467330028939693481001494, −5.69075308204608178241887311036, −4.86461386710839093380957240894, −2.98824434028231518830236954831, −2.18442979252484930721869252193, −0.562114427317718191586095097101, 0.99165098167171404645569686905, 2.59981816916236480995167982466, 3.11190331663072296794401512002, 4.454251933269767345328235617742, 5.06416861688442712681086038592, 5.99496735911087054260075251870, 7.40303514167256580389479378722, 8.677237402210497835951889635, 9.703322431218347235780079968096, 9.92506327690948810081537173944, 10.59972579206881567067267431040, 11.69277798692807514936322933521, 12.56358654868496574438144909302, 13.529624400668075079136587603659, 13.9948982346200232345889811114, 15.21572520268997169747900450424, 16.22594344172170645484266404258, 17.04487567820771988914532019350, 17.62310849280248606749585065899, 18.4568436815331909232984054760, 19.746345477647530312772956833564, 20.1518903842160929797628775515, 21.01833313200656998418867085265, 21.73734060202684807891830201198, 22.176426870417310746510376251870

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