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Properties

Label 1-731-731.328-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.0886 + 0.996i$
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  + (−0.330 + 0.943i)2-s + (−0.745 + 0.666i)3-s + (−0.781 − 0.623i)4-s + (−0.578 + 0.815i)5-s + (−0.382 − 0.923i)6-s + (0.923 − 0.382i)7-s + (0.846 − 0.532i)8-s + (0.111 − 0.993i)9-s + (−0.578 − 0.815i)10-s + (−0.483 + 0.875i)11-s + (0.998 − 0.0560i)12-s + (−0.974 − 0.222i)13-s + (0.0560 + 0.998i)14-s + (−0.111 − 0.993i)15-s + (0.222 + 0.974i)16-s + ⋯
L(s)  = 1  + (−0.330 + 0.943i)2-s + (−0.745 + 0.666i)3-s + (−0.781 − 0.623i)4-s + (−0.578 + 0.815i)5-s + (−0.382 − 0.923i)6-s + (0.923 − 0.382i)7-s + (0.846 − 0.532i)8-s + (0.111 − 0.993i)9-s + (−0.578 − 0.815i)10-s + (−0.483 + 0.875i)11-s + (0.998 − 0.0560i)12-s + (−0.974 − 0.222i)13-s + (0.0560 + 0.998i)14-s + (−0.111 − 0.993i)15-s + (0.222 + 0.974i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0886 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0886 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.0886 + 0.996i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.0886 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4822286969 + 0.5270711988i\)
\(L(\frac12)\) \(\approx\) \(0.4822286969 + 0.5270711988i\)
\(L(1)\) \(\approx\) \(0.5184173096 + 0.3853083429i\)
\(L(1)\) \(\approx\) \(0.5184173096 + 0.3853083429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.330 + 0.943i)T \)
3 \( 1 + (-0.745 + 0.666i)T \)
5 \( 1 + (-0.578 + 0.815i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (-0.483 + 0.875i)T \)
13 \( 1 + (-0.974 - 0.222i)T \)
19 \( 1 + (-0.111 - 0.993i)T \)
23 \( 1 + (0.483 - 0.875i)T \)
29 \( 1 + (0.998 - 0.0560i)T \)
31 \( 1 + (0.0560 + 0.998i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (0.666 - 0.745i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (0.846 + 0.532i)T \)
59 \( 1 + (0.532 - 0.846i)T \)
61 \( 1 + (0.998 + 0.0560i)T \)
67 \( 1 + (-0.623 + 0.781i)T \)
71 \( 1 + (-0.483 - 0.875i)T \)
73 \( 1 + (-0.578 + 0.815i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (-0.943 + 0.330i)T \)
89 \( 1 + (0.433 - 0.900i)T \)
97 \( 1 + (0.276 + 0.960i)T \)
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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

−22.12310048408899748131459720230, −21.4098070931855376710535600047, −20.75878198458008110784027823454, −19.691744553838972887519048237941, −19.07553079985549777775432959073, −18.404674749739388904467613897934, −17.510544797515185843696970720665, −16.866297775442124736497345491494, −16.169610097346250914010976973334, −14.82816745101372466217446427713, −13.66233927369960308421391163150, −12.99553678387219554806480069276, −11.97394475710240596013770427446, −11.76915973000946238870841133696, −10.915306425855174454805970517701, −9.912097580927388477506045691889, −8.69490119995026268211110351910, −8.03799143971936028938485343901, −7.41785451731461595410695516884, −5.72138417206462257184794561576, −5.00662333397895404860595592958, −4.192554887350660393231934730993, −2.736105179317260818230025884587, −1.668884710338934750660432820060, −0.75593913902465278895211381011, 0.69966590489409071898645854868, 2.55964456897103832092669972949, 4.19141458757311942315446173215, 4.68829832524487926637369956626, 5.52105477591091336805418501226, 6.88597080289855990271652022525, 7.19220538978637016304831186278, 8.24988724894549647168131919109, 9.33226320013943556685481520960, 10.49839862454020221916020684569, 10.60782015198871001371555800982, 11.822815913840107317486367225683, 12.78078294223627428692523873958, 14.27130473421807919458063709662, 14.7128536005586431599366294533, 15.46651894874359210359925732315, 16.06170369565865617988443459509, 17.23221338010854693643897547692, 17.6568785461762924161722650827, 18.25605940301545503379358797092, 19.38311965123484751531641887525, 20.22676297365139750735393718844, 21.363815513263572290887677291687, 22.19621813197149692412209741666, 22.907750943428043441703462392341

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