↑

Properties

Label 1-647-647.53-r0-0-0
Degree $1$
Conductor $647$
Sign $-0.260 - 0.965i$
Analytic cond. $3.00465$
Root an. cond. $3.00465$
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  + (−0.982 − 0.183i)2-s + (−0.602 + 0.798i)3-s + (0.932 + 0.361i)4-s + (−0.273 − 0.961i)5-s + (0.739 − 0.673i)6-s + (−0.850 + 0.526i)7-s + (−0.850 − 0.526i)8-s + (−0.273 − 0.961i)9-s + (0.0922 + 0.995i)10-s + (−0.273 + 0.961i)11-s + (−0.850 + 0.526i)12-s + (−0.850 + 0.526i)13-s + (0.932 − 0.361i)14-s + (0.932 + 0.361i)15-s + (0.739 + 0.673i)16-s + (−0.602 + 0.798i)17-s + ⋯
L(s)  = 1  + (−0.982 − 0.183i)2-s + (−0.602 + 0.798i)3-s + (0.932 + 0.361i)4-s + (−0.273 − 0.961i)5-s + (0.739 − 0.673i)6-s + (−0.850 + 0.526i)7-s + (−0.850 − 0.526i)8-s + (−0.273 − 0.961i)9-s + (0.0922 + 0.995i)10-s + (−0.273 + 0.961i)11-s + (−0.850 + 0.526i)12-s + (−0.850 + 0.526i)13-s + (0.932 − 0.361i)14-s + (0.932 + 0.361i)15-s + (0.739 + 0.673i)16-s + (−0.602 + 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(647\)
Sign: $-0.260 - 0.965i$
Analytic conductor: \(3.00465\)
Root analytic conductor: \(3.00465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{647} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 647,\ (0:\ ),\ -0.260 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07594457982 - 0.09910610226i\)
\(L(\frac12)\) \(\approx\) \(0.07594457982 - 0.09910610226i\)
\(L(1)\) \(\approx\) \(0.3956696959 + 0.05273466557i\)
\(L(1)\) \(\approx\) \(0.3956696959 + 0.05273466557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad647 \( 1 \)
good2 \( 1 + (-0.982 - 0.183i)T \)
3 \( 1 + (-0.602 + 0.798i)T \)
5 \( 1 + (-0.273 - 0.961i)T \)
7 \( 1 + (-0.850 + 0.526i)T \)
11 \( 1 + (-0.273 + 0.961i)T \)
13 \( 1 + (-0.850 + 0.526i)T \)
17 \( 1 + (-0.602 + 0.798i)T \)
19 \( 1 + (0.739 + 0.673i)T \)
23 \( 1 + (0.739 - 0.673i)T \)
29 \( 1 + (0.739 - 0.673i)T \)
31 \( 1 + (-0.273 + 0.961i)T \)
37 \( 1 + (0.739 - 0.673i)T \)
41 \( 1 + (-0.850 - 0.526i)T \)
43 \( 1 + (-0.982 + 0.183i)T \)
47 \( 1 + (-0.602 + 0.798i)T \)
53 \( 1 + (0.739 - 0.673i)T \)
59 \( 1 + (0.0922 + 0.995i)T \)
61 \( 1 + (-0.602 - 0.798i)T \)
67 \( 1 + (-0.602 - 0.798i)T \)
71 \( 1 + (0.739 - 0.673i)T \)
73 \( 1 + (-0.982 - 0.183i)T \)
79 \( 1 + (-0.982 + 0.183i)T \)
83 \( 1 + (0.0922 - 0.995i)T \)
89 \( 1 + (-0.982 - 0.183i)T \)
97 \( 1 + (0.0922 + 0.995i)T \)
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

−23.30745290880277819831135244182, −22.362492452359819185771151669525, −21.72748063832736429374834213045, −19.99366020597130388501216717621, −19.7481356943633495056825449980, −18.75538621039396587137899837979, −18.33175776941298234193530665288, −17.447676630661075516868963370530, −16.665273466061996547505447281688, −15.907846306734703676560345724430, −15.04741269664973394349988625608, −13.823048908490127502522001773229, −13.1270273990591979870188023978, −11.75965818376856326554826560533, −11.30128229022599433623487721678, −10.42079809954342692074964741049, −9.677828233217247760414309651544, −8.39163035776527906862359962601, −7.33641804587717133809576390817, −7.02191935087669949700371337814, −6.15791829126927247676862815950, −5.14605748220185159325070500103, −3.12572497053514325019621421657, −2.60804566813227764061021210770, −0.94842431876437467007680235685, 0.114429326772743602512883525812, 1.69028245161263202301730479771, 2.98884789430625000208294577482, 4.17938369378095094344255428043, 5.13268867415326869916861631542, 6.22987434474668695503234592550, 7.13492513099957030224301333603, 8.383512975330772749342565310420, 9.19788380669407717109267251961, 9.78612747274491455959242191050, 10.50958112534359989955833186837, 11.762021731035280041434765599126, 12.250323317307459888630891810719, 12.92077700237967893161029956198, 14.8120859644292670450859607179, 15.50642931642832701797829423287, 16.22633573782968919124728737482, 16.80095389593775899891045547897, 17.539998107309751843048600173039, 18.41282904456744839617745249665, 19.56779892410121483980386053374, 20.01122370089700599548381508038, 20.992506717946742927490861858394, 21.566772751018601440864612228, 22.54255696093246736289588754054

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