↑

Properties

Label 1-709-709.145-r0-0-0
Degree $1$
Conductor $709$
Sign $0.848 + 0.528i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.949 − 0.314i)2-s + (−0.0266 − 0.999i)3-s + (0.802 + 0.596i)4-s + (−0.697 + 0.716i)5-s + (−0.288 + 0.957i)6-s + (−0.617 − 0.786i)7-s + (−0.574 − 0.818i)8-s + (−0.998 + 0.0532i)9-s + (0.887 − 0.461i)10-s + (0.910 + 0.413i)11-s + (0.574 − 0.818i)12-s + (0.617 − 0.786i)13-s + (0.339 + 0.940i)14-s + (0.734 + 0.678i)15-s + (0.288 + 0.957i)16-s + (−0.910 + 0.413i)17-s + ⋯
L(s)  = 1  + (−0.949 − 0.314i)2-s + (−0.0266 − 0.999i)3-s + (0.802 + 0.596i)4-s + (−0.697 + 0.716i)5-s + (−0.288 + 0.957i)6-s + (−0.617 − 0.786i)7-s + (−0.574 − 0.818i)8-s + (−0.998 + 0.0532i)9-s + (0.887 − 0.461i)10-s + (0.910 + 0.413i)11-s + (0.574 − 0.818i)12-s + (0.617 − 0.786i)13-s + (0.339 + 0.940i)14-s + (0.734 + 0.678i)15-s + (0.288 + 0.957i)16-s + (−0.910 + 0.413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.848 + 0.528i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.848 + 0.528i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4174488779 + 0.1193107075i\)
\(L(\frac12)\) \(\approx\) \(0.4174488779 + 0.1193107075i\)
\(L(1)\) \(\approx\) \(0.5112124678 - 0.1352259474i\)
\(L(1)\) \(\approx\) \(0.5112124678 - 0.1352259474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.949 - 0.314i)T \)
3 \( 1 + (-0.0266 - 0.999i)T \)
5 \( 1 + (-0.697 + 0.716i)T \)
7 \( 1 + (-0.617 - 0.786i)T \)
11 \( 1 + (0.910 + 0.413i)T \)
13 \( 1 + (0.617 - 0.786i)T \)
17 \( 1 + (-0.910 + 0.413i)T \)
19 \( 1 + (0.574 + 0.818i)T \)
23 \( 1 + (-0.994 + 0.106i)T \)
29 \( 1 + (-0.964 + 0.263i)T \)
31 \( 1 + (-0.734 - 0.678i)T \)
37 \( 1 + (0.617 + 0.786i)T \)
41 \( 1 + (-0.861 + 0.507i)T \)
43 \( 1 + (0.388 + 0.921i)T \)
47 \( 1 + (0.994 - 0.106i)T \)
53 \( 1 + (-0.734 - 0.678i)T \)
59 \( 1 + (0.574 + 0.818i)T \)
61 \( 1 + (-0.658 - 0.752i)T \)
67 \( 1 + (-0.437 - 0.899i)T \)
71 \( 1 + (0.887 + 0.461i)T \)
73 \( 1 + (-0.977 + 0.211i)T \)
79 \( 1 + (0.931 + 0.364i)T \)
83 \( 1 + (0.237 + 0.971i)T \)
89 \( 1 + (0.833 + 0.552i)T \)
97 \( 1 + (0.987 + 0.159i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.34378294276162541983183342300, −21.77762866061245438117796962411, −20.6124839020836809401279637077, −20.04473545703404870766338999270, −19.377396283463154283497601646778, −18.526459509807531310157740180966, −17.45391321087462721024296347393, −16.59232016782482467111937335803, −15.972636617310391901782569149782, −15.65017335707318299357153034822, −14.67609119465773077440833746008, −13.6323930466518671621739985699, −12.09963073530151161619966757809, −11.54310256048382091575084538435, −10.83164419829254021945099636243, −9.48995210223766203834794857851, −9.00689059874371021846939504754, −8.664850187874006524332474143316, −7.30740271923558623386384609371, −6.22746385602789615199818273420, −5.44104550189784744099087775721, −4.26871099806419095842889566911, −3.31841143722486483229882606754, −1.98284486055317798672892008942, −0.34177838842370256870700087967, 1.00846776045585858483575221123, 2.08357916611369952649881360594, 3.32056008607461510100829206869, 3.89575354027926254291572072161, 6.13714072166499441425013754426, 6.60918270771143195598268487288, 7.59097383064854139967936902218, 7.98433922701792677656995283746, 9.17476843718867649271476423865, 10.18469312958713074242591370817, 11.04548160202792299673768037341, 11.69315706355173563352336838033, 12.58134427916213254488387584822, 13.357969551031927805408629253462, 14.476959489697793090351451462805, 15.40478872668670307044627997539, 16.417943728133434234678422602375, 17.13724801696578930654505449368, 18.07155031389164984017769866, 18.532102888081861369122275759712, 19.4565028020275495201246864382, 20.06887059193114277058705458105, 20.37962066017854616627990079765, 22.23676958002955661611894591802, 22.51987373284869909538249010371

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