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Properties

Label 1-731-731.211-r0-0-0
Degree $1$
Conductor $731$
Sign $0.775 - 0.631i$
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  + (0.532 − 0.846i)2-s + (−0.985 + 0.167i)3-s + (−0.433 − 0.900i)4-s + (0.875 + 0.483i)5-s + (−0.382 + 0.923i)6-s + (0.923 + 0.382i)7-s + (−0.993 − 0.111i)8-s + (0.943 − 0.330i)9-s + (0.875 − 0.483i)10-s + (−0.745 + 0.666i)11-s + (0.578 + 0.815i)12-s + (0.781 − 0.623i)13-s + (0.815 − 0.578i)14-s + (−0.943 − 0.330i)15-s + (−0.623 + 0.781i)16-s + ⋯
L(s)  = 1  + (0.532 − 0.846i)2-s + (−0.985 + 0.167i)3-s + (−0.433 − 0.900i)4-s + (0.875 + 0.483i)5-s + (−0.382 + 0.923i)6-s + (0.923 + 0.382i)7-s + (−0.993 − 0.111i)8-s + (0.943 − 0.330i)9-s + (0.875 − 0.483i)10-s + (−0.745 + 0.666i)11-s + (0.578 + 0.815i)12-s + (0.781 − 0.623i)13-s + (0.815 − 0.578i)14-s + (−0.943 − 0.330i)15-s + (−0.623 + 0.781i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.775 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.574942722 - 0.5597558930i\)
\(L(\frac12)\) \(\approx\) \(1.574942722 - 0.5597558930i\)
\(L(1)\) \(\approx\) \(1.192070394 - 0.3872785382i\)
\(L(1)\) \(\approx\) \(1.192070394 - 0.3872785382i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.532 - 0.846i)T \)
3 \( 1 + (-0.985 + 0.167i)T \)
5 \( 1 + (0.875 + 0.483i)T \)
7 \( 1 + (0.923 + 0.382i)T \)
11 \( 1 + (-0.745 + 0.666i)T \)
13 \( 1 + (0.781 - 0.623i)T \)
19 \( 1 + (-0.943 - 0.330i)T \)
23 \( 1 + (0.745 - 0.666i)T \)
29 \( 1 + (0.578 + 0.815i)T \)
31 \( 1 + (0.815 - 0.578i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.167 + 0.985i)T \)
47 \( 1 + (0.433 + 0.900i)T \)
53 \( 1 + (-0.993 + 0.111i)T \)
59 \( 1 + (-0.111 - 0.993i)T \)
61 \( 1 + (0.578 - 0.815i)T \)
67 \( 1 + (0.900 - 0.433i)T \)
71 \( 1 + (-0.745 - 0.666i)T \)
73 \( 1 + (0.875 + 0.483i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.846 + 0.532i)T \)
89 \( 1 + (0.974 + 0.222i)T \)
97 \( 1 + (-0.998 - 0.0560i)T \)
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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

−22.92130941905161128295135699372, −21.63340230332181074338883071278, −21.28578066463628158537821452116, −20.81088623382442981074213524729, −19.01789408080269708919908242215, −18.16462127578359661317688069996, −17.476724113160175128874489833987, −16.96109820574947154280666807999, −16.20441059202676528515632850026, −15.4354693713761773394711854990, −14.198826894228856413579375984198, −13.55417941119272495844916530770, −12.95743960324472678992515227238, −11.95478859254485983774633468879, −11.07687784125936591813058076075, −10.220907472930426472791060100434, −8.87964664050185037416885291955, −8.15962175729786887539216075629, −7.094927112365404170961330909944, −6.18262191673861625985184694932, −5.53055265682924111081569268053, −4.78945147784829349499745158634, −3.95014113138681904515679848326, −2.24019413734836887948839231744, −0.960844698138626972471013367324, 1.10294908942977381445570439194, 2.088072113581145146053520849262, 3.05740376935825113711483506222, 4.60157493370623707739231216070, 5.00647057933297948505888622585, 5.99872538969392175620875523486, 6.6499895980946032261357987065, 8.19040337419261216609503903879, 9.38981573042081012324255101825, 10.309188912504657570677810292561, 10.83055725003018110264330141276, 11.435458235560694132098595528366, 12.63815565325852641732444789569, 13.021544690143493701368504593818, 14.12827555966318488974223254027, 15.06517499966854166908166606157, 15.557541275810480604739171287616, 17.0765814886419394888174527011, 17.78976537999062517145729049963, 18.31710187959432502068136477490, 18.94343413172033359564865633884, 20.562904592422072016898191695595, 20.87504017013117230054540603997, 21.69321600769902957512131497287, 22.24105136996082419258133708636

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