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Properties

Label 1-731-731.125-r0-0-0
Degree $1$
Conductor $731$
Sign $0.989 + 0.143i$
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 + (−0.167 − 0.985i)3-s + (−0.433 − 0.900i)4-s + (−0.483 + 0.875i)5-s + (0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s + (0.993 + 0.111i)8-s + (−0.943 + 0.330i)9-s + (−0.483 − 0.875i)10-s + (0.666 + 0.745i)11-s + (−0.815 + 0.578i)12-s + (0.781 − 0.623i)13-s + (0.578 + 0.815i)14-s + (0.943 + 0.330i)15-s + (−0.623 + 0.781i)16-s + ⋯
L(s)  = 1  + (−0.532 + 0.846i)2-s + (−0.167 − 0.985i)3-s + (−0.433 − 0.900i)4-s + (−0.483 + 0.875i)5-s + (0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s + (0.993 + 0.111i)8-s + (−0.943 + 0.330i)9-s + (−0.483 − 0.875i)10-s + (0.666 + 0.745i)11-s + (−0.815 + 0.578i)12-s + (0.781 − 0.623i)13-s + (0.578 + 0.815i)14-s + (0.943 + 0.330i)15-s + (−0.623 + 0.781i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.989 + 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9363855243 + 0.06769799193i\)
\(L(\frac12)\) \(\approx\) \(0.9363855243 + 0.06769799193i\)
\(L(1)\) \(\approx\) \(0.7711822479 + 0.07465845173i\)
\(L(1)\) \(\approx\) \(0.7711822479 + 0.07465845173i\)

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 \( 1 + (-0.167 - 0.985i)T \)
5 \( 1 + (-0.483 + 0.875i)T \)
7 \( 1 + (0.382 - 0.923i)T \)
11 \( 1 + (0.666 + 0.745i)T \)
13 \( 1 + (0.781 - 0.623i)T \)
19 \( 1 + (0.943 + 0.330i)T \)
23 \( 1 + (-0.666 - 0.745i)T \)
29 \( 1 + (-0.815 + 0.578i)T \)
31 \( 1 + (0.578 + 0.815i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (0.985 + 0.167i)T \)
47 \( 1 + (0.433 + 0.900i)T \)
53 \( 1 + (0.993 - 0.111i)T \)
59 \( 1 + (0.111 + 0.993i)T \)
61 \( 1 + (-0.815 - 0.578i)T \)
67 \( 1 + (0.900 - 0.433i)T \)
71 \( 1 + (0.666 - 0.745i)T \)
73 \( 1 + (-0.483 + 0.875i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (0.846 - 0.532i)T \)
89 \( 1 + (0.974 + 0.222i)T \)
97 \( 1 + (0.0560 - 0.998i)T \)
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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

−22.19130266408232992751519037638, −21.46098480370997529263419356004, −20.94667797073615770186140791782, −20.19067292800811081355836104241, −19.3952264344028427294235254386, −18.59102598421009064849883426694, −17.56558099691064948808174331043, −16.834747640971743876151002572966, −16.0426464362644098910085428513, −15.51866337982442739348687993927, −14.17952193738935402201448728436, −13.3540527880966968421138651004, −12.03017301030463653284961052154, −11.63158249913167398345901388184, −11.09827836894157264150637469422, −9.773687471108748968625755127601, −9.04577256283931039937958284101, −8.66934517409078282410558848070, −7.70097773364330367316622700006, −5.95872673075295533836576812752, −5.10107329111014652297224580982, −4.03021873979163782605863549631, −3.51775411182891420558732878377, −2.13057540486712097661525674636, −0.85700805818253751695990561302, 0.8427535606885429083083553675, 1.81477074816257125484121552700, 3.382112038924452096121271210146, 4.5323551157068180549706783293, 5.79660551372370068870063835738, 6.63080794565821570986716175083, 7.317046179148936392416248497659, 7.83068935858709419041966337526, 8.74757321220777086198216958761, 10.099849622475358225384391191615, 10.77755316660091988589855677853, 11.62724868601994047635089798939, 12.71016834334701956290549572728, 13.96421396222584552661801554817, 14.16861686695408741725973814383, 15.1415179601929031274186155117, 16.12610424564979424513666144252, 17.00010840328676972111837551477, 17.92034525077115441809367172070, 18.13823512276012135174705226136, 19.11378646427857880065373929875, 19.91604095455155022843382660466, 20.44343866707147243468190949169, 22.341154109930270196276565441049, 22.87958797857874172450129455570

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