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Properties

Label 1-731-731.183-r0-0-0
Degree $1$
Conductor $731$
Sign $0.0830 - 0.996i$
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  + (0.222 − 0.974i)2-s + (0.974 − 0.222i)3-s + (−0.900 − 0.433i)4-s + (0.781 + 0.623i)5-s − i·6-s − i·7-s + (−0.623 + 0.781i)8-s + (0.900 − 0.433i)9-s + (0.781 − 0.623i)10-s + (0.433 + 0.900i)11-s + (−0.974 − 0.222i)12-s + (0.623 − 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.900 + 0.433i)15-s + (0.623 + 0.781i)16-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (0.974 − 0.222i)3-s + (−0.900 − 0.433i)4-s + (0.781 + 0.623i)5-s − i·6-s − i·7-s + (−0.623 + 0.781i)8-s + (0.900 − 0.433i)9-s + (0.781 − 0.623i)10-s + (0.433 + 0.900i)11-s + (−0.974 − 0.222i)12-s + (0.623 − 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.900 + 0.433i)15-s + (0.623 + 0.781i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0830 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0830 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.0830 - 0.996i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.0830 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.883330799 - 1.732836216i\)
\(L(\frac12)\) \(\approx\) \(1.883330799 - 1.732836216i\)
\(L(1)\) \(\approx\) \(1.512944356 - 0.9221817799i\)
\(L(1)\) \(\approx\) \(1.512944356 - 0.9221817799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.222 - 0.974i)T \)
3 \( 1 + (0.974 - 0.222i)T \)
5 \( 1 + (0.781 + 0.623i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.433 + 0.900i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (0.433 + 0.900i)T \)
29 \( 1 + (-0.974 - 0.222i)T \)
31 \( 1 + (-0.974 - 0.222i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (-0.974 + 0.222i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (0.433 - 0.900i)T \)
73 \( 1 + (0.781 + 0.623i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.433 - 0.900i)T \)
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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

−22.561375615975214554698005121793, −21.749677637798546253657391846690, −21.35834862776724950100885178686, −20.52092292623726986805958847958, −19.31711326069441415055959906113, −18.52025758123705306825837531747, −17.926605595165720153376267181379, −16.52934546603115783603335109275, −16.3018012412578530292514363575, −15.342368346174321438495619977351, −14.37539095699763119166654785433, −13.95066900803166801172903097804, −13.07468372148699721042680983799, −12.39680224329955732085036011870, −11.03090970368944373598941873784, −9.4977657966080140431911817441, −9.05579868902768124072148908855, −8.65117476731106370992670411652, −7.553347205084361496159435392146, −6.40266344077339834417024143316, −5.63247310344758406454708893405, −4.73624105491995388898461015108, −3.695457960230685407395129146121, −2.66650671911132545350810087414, −1.37730975024047370308597347237, 1.29382788800565180642366267703, 1.893249143419630343626102208868, 3.21366467749704850496729712593, 3.603458447929954208735671090260, 4.8180834913780959671443447493, 6.03487972843363628850603318549, 7.1898718037848146522603718266, 7.959170364703442539537827249158, 9.365457165170778883673489424424, 9.70332111494429963298376643286, 10.55575156778604433095357006935, 11.41113812234674684485022090052, 12.71717210855165713360251728069, 13.291012358765931982976917376, 13.9317263513043475102762452254, 14.637550232426950812288865203497, 15.338732725560092094879862510619, 16.912126731839529492603082253198, 17.9049211532008751885399754188, 18.282127527208558598891962794427, 19.320962128423060891646409270636, 20.08049359876389433081605191267, 20.60953522408997339660662019764, 21.230123412955338591555453937103, 22.35043259345273257555359611072

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