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Properties

Label 1-731-731.112-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.401 + 0.915i$
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  + (−0.943 − 0.330i)2-s + (−0.836 − 0.547i)3-s + (0.781 + 0.623i)4-s + (0.937 + 0.347i)5-s + (0.608 + 0.793i)6-s + (0.130 + 0.991i)7-s + (−0.532 − 0.846i)8-s + (0.399 + 0.916i)9-s + (−0.770 − 0.638i)10-s + (−0.960 + 0.276i)11-s + (−0.312 − 0.949i)12-s + (−0.294 − 0.955i)13-s + (0.204 − 0.978i)14-s + (−0.593 − 0.804i)15-s + (0.222 + 0.974i)16-s + ⋯
L(s)  = 1  + (−0.943 − 0.330i)2-s + (−0.836 − 0.547i)3-s + (0.781 + 0.623i)4-s + (0.937 + 0.347i)5-s + (0.608 + 0.793i)6-s + (0.130 + 0.991i)7-s + (−0.532 − 0.846i)8-s + (0.399 + 0.916i)9-s + (−0.770 − 0.638i)10-s + (−0.960 + 0.276i)11-s + (−0.312 − 0.949i)12-s + (−0.294 − 0.955i)13-s + (0.204 − 0.978i)14-s + (−0.593 − 0.804i)15-s + (0.222 + 0.974i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.401 + 0.915i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.401 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2191071553 + 0.3354800580i\)
\(L(\frac12)\) \(\approx\) \(0.2191071553 + 0.3354800580i\)
\(L(1)\) \(\approx\) \(0.5327081958 + 0.03095063555i\)
\(L(1)\) \(\approx\) \(0.5327081958 + 0.03095063555i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.943 - 0.330i)T \)
3 \( 1 + (-0.836 - 0.547i)T \)
5 \( 1 + (0.937 + 0.347i)T \)
7 \( 1 + (0.130 + 0.991i)T \)
11 \( 1 + (-0.960 + 0.276i)T \)
13 \( 1 + (-0.294 - 0.955i)T \)
19 \( 1 + (-0.399 + 0.916i)T \)
23 \( 1 + (-0.240 + 0.970i)T \)
29 \( 1 + (0.978 + 0.204i)T \)
31 \( 1 + (-0.949 + 0.312i)T \)
37 \( 1 + (0.991 + 0.130i)T \)
41 \( 1 + (-0.998 + 0.0560i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (-0.467 - 0.884i)T \)
59 \( 1 + (-0.846 - 0.532i)T \)
61 \( 1 + (-0.312 + 0.949i)T \)
67 \( 1 + (-0.365 - 0.930i)T \)
71 \( 1 + (0.240 + 0.970i)T \)
73 \( 1 + (-0.770 + 0.638i)T \)
79 \( 1 + (0.991 - 0.130i)T \)
83 \( 1 + (0.982 + 0.185i)T \)
89 \( 1 + (-0.563 - 0.826i)T \)
97 \( 1 + (-0.875 - 0.483i)T \)
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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

−22.103032391021957023438133824434, −21.316620280027703760195611777688, −20.696815852490616743054872495010, −19.91947057623351746786611291909, −18.71596373470945102974534251867, −17.95238294849682705919628382705, −17.34173682879449431570416490501, −16.556307637156854858364587815317, −16.27050112672545636238784052004, −15.11225442342411361311238611372, −14.18400726387277361617607894528, −13.22990755724505507820882618360, −12.11167147625311974344626020551, −10.942742353212585654382324053130, −10.586500779883673389182615197430, −9.715502943535416410458741977269, −9.05366063810027830152980670721, −7.91161699677951513185475249767, −6.76684603995981495999371764859, −6.23808490154184004961032394671, −5.112649039210478578230418944311, −4.44703872227612005856418372809, −2.69332639798551922008151558565, −1.46838482615378792821708886029, −0.28638806486588797176154032187, 1.47829664968317333548679066151, 2.21826773769126696995747893684, 3.108898862459561466665287093166, 5.118555910123375467999548325188, 5.78590893247693646367639431703, 6.61722416524658241014857958986, 7.67297025913576316236516089841, 8.35246321594770268520976581035, 9.63604920232525934335714381546, 10.26960893754577851999168993302, 10.968558848627830114442404136920, 11.96973697034036083210968532866, 12.67777469387177443382383984977, 13.31572288065089106800807161130, 14.77289837027904027593807869851, 15.65328905913628419983310238045, 16.52766095559240399690289293241, 17.44245859018757178330599794017, 18.07795840994404201648117056105, 18.35747367582281535567443121015, 19.23410810423030916698618269826, 20.27893498999323523558560318834, 21.374325733500491202673467595907, 21.708202867104785422768397130684, 22.622575997375223396685861065

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