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Properties

Label 1-709-709.147-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.641 + 0.766i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (−0.769 + 0.638i)2-s + (0.949 + 0.314i)3-s + (0.185 − 0.982i)4-s + (−0.987 − 0.159i)5-s + (−0.931 + 0.364i)6-s + (−0.132 + 0.991i)7-s + (0.484 + 0.874i)8-s + (0.802 + 0.596i)9-s + (0.861 − 0.507i)10-s + (0.388 + 0.921i)11-s + (0.484 − 0.874i)12-s + (−0.132 − 0.991i)13-s + (−0.530 − 0.847i)14-s + (−0.887 − 0.461i)15-s + (−0.931 − 0.364i)16-s + (0.388 − 0.921i)17-s + ⋯
L(s)  = 1  + (−0.769 + 0.638i)2-s + (0.949 + 0.314i)3-s + (0.185 − 0.982i)4-s + (−0.987 − 0.159i)5-s + (−0.931 + 0.364i)6-s + (−0.132 + 0.991i)7-s + (0.484 + 0.874i)8-s + (0.802 + 0.596i)9-s + (0.861 − 0.507i)10-s + (0.388 + 0.921i)11-s + (0.484 − 0.874i)12-s + (−0.132 − 0.991i)13-s + (−0.530 − 0.847i)14-s + (−0.887 − 0.461i)15-s + (−0.931 − 0.364i)16-s + (0.388 − 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.641 + 0.766i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.641 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4348225923 + 0.9311221839i\)
\(L(\frac12)\) \(\approx\) \(0.4348225923 + 0.9311221839i\)
\(L(1)\) \(\approx\) \(0.7364170048 + 0.4641859699i\)
\(L(1)\) \(\approx\) \(0.7364170048 + 0.4641859699i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.769 + 0.638i)T \)
3 \( 1 + (0.949 + 0.314i)T \)
5 \( 1 + (-0.987 - 0.159i)T \)
7 \( 1 + (-0.132 + 0.991i)T \)
11 \( 1 + (0.388 + 0.921i)T \)
13 \( 1 + (-0.132 - 0.991i)T \)
17 \( 1 + (0.388 - 0.921i)T \)
19 \( 1 + (0.484 + 0.874i)T \)
23 \( 1 + (0.288 + 0.957i)T \)
29 \( 1 + (-0.998 - 0.0532i)T \)
31 \( 1 + (-0.887 - 0.461i)T \)
37 \( 1 + (-0.132 + 0.991i)T \)
41 \( 1 + (0.994 + 0.106i)T \)
43 \( 1 + (0.0797 - 0.996i)T \)
47 \( 1 + (0.288 + 0.957i)T \)
53 \( 1 + (-0.887 - 0.461i)T \)
59 \( 1 + (0.484 + 0.874i)T \)
61 \( 1 + (-0.697 + 0.716i)T \)
67 \( 1 + (0.658 - 0.752i)T \)
71 \( 1 + (0.861 + 0.507i)T \)
73 \( 1 + (-0.833 + 0.552i)T \)
79 \( 1 + (-0.237 + 0.971i)T \)
83 \( 1 + (-0.964 + 0.263i)T \)
89 \( 1 + (0.734 - 0.678i)T \)
97 \( 1 + (-0.339 - 0.940i)T \)
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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

−22.02506699295350188485461910663, −21.29001030546018431920556095464, −20.29677823483064774781335033606, −19.83402871564588213432477765568, −19.09269707384116510889728104494, −18.75007275173644982219747991529, −17.560097941457936106692315023471, −16.51151882095193057601376518385, −16.07763177006213651596737434206, −14.76491808157173917057910106935, −14.046188671421869157471963525631, −13.04535430067984753790149685450, −12.34429751492615955047213168972, −11.22461763462228575090289944483, −10.74331729635870810157075715084, −9.4851727559997904894512806219, −8.83037377432719783909416980082, −7.97277125101646157973128153232, −7.25846757561990192316592613284, −6.59700035153097102030590574509, −4.33736868690747787913867499856, −3.72194123008800020572199381017, −3.01316795513124424887567908694, −1.70445170039486317429031549735, −0.61383852779993615431088589294, 1.378342228112604562841607502776, 2.59091848440204449351158969437, 3.64230160695805771670621216042, 4.88431166221918401467085691832, 5.68095601502159324862383323937, 7.26685007643342748355677361828, 7.62571494496142571067448464282, 8.49931540610616207299364118414, 9.38629141989556687808946341407, 9.83590053872953872416887509406, 11.08826720132490354166910524410, 12.058489317119028173636945304908, 12.98723004227977874334124822038, 14.313609278066184003008088619851, 14.96064453819254385803854443712, 15.5285535482860317353847729011, 16.071245040318984654881080134214, 17.086988114027833650624697307194, 18.30275924287815303732837066513, 18.75766709623179785617806775961, 19.627670196211629546231522214377, 20.27544167328242064155383451397, 20.84776659745190188471386294502, 22.40358529761657000736213487753, 22.81270041338690082929556203314

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