↑

Properties

Label 1-65-65.19-r1-0-0
Degree $1$
Conductor $65$
Sign $0.999 - 0.0386i$
Analytic cond. $6.98522$
Root an. cond. $6.98522$
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  + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s + 12-s + 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + i·18-s + (0.866 + 0.5i)19-s + i·21-s + (0.5 − 0.866i)22-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s + 12-s + 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + i·18-s + (0.866 + 0.5i)19-s + i·21-s + (0.5 − 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.999 - 0.0386i$
Analytic conductor: \(6.98522\)
Root analytic conductor: \(6.98522\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (1:\ ),\ 0.999 - 0.0386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.229871953 - 0.06236648643i\)
\(L(\frac12)\) \(\approx\) \(3.229871953 - 0.06236648643i\)
\(L(1)\) \(\approx\) \(2.127590830 - 0.08219512004i\)
\(L(1)\) \(\approx\) \(2.127590830 - 0.08219512004i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 - T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + iT \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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

−31.7639326909223845696210058602, −30.767610212255018015117741125461, −30.1829745371517889549102178652, −29.11490419086941040592122418951, −27.24752736520471318237995605511, −26.03659075100815254542416170610, −24.96309559605233110051780295190, −24.2120860481640623517422691084, −23.27586634342688000590543608409, −22.05761290327765309343038989786, −20.55811714297253722690938313302, −19.90307835242509907325962540465, −18.02354474794334512827073804404, −17.21980893069045637825864970028, −15.5679349758644484219917551413, −14.27148067264854690316621445400, −13.748789912837323678936680955980, −12.310697547680493609380411996016, −11.35614467962968662655698996591, −8.9856156604319244739583702111, −7.57338318890097386351541433815, −6.81230050636589987267820678736, −5.07022885917931684307300402042, −3.495676542980613625701408188492, −1.73976267148493061907840072907, 1.94175854019992234704097928409, 3.54911886836600836465720757924, 4.70891087949584517460125419143, 6.04254990676869338295814969607, 8.24982868243246652068410258114, 9.641452614227603836380787844990, 10.962938494340186887749990479268, 11.92465649194042800368242182674, 13.63969006582202510105756777805, 14.59754515436947403787797043464, 15.40551681868759016196967126985, 16.794796069745644959386736994133, 18.68639716934038246828226994493, 19.921489052576328006310731106609, 20.82965350778209242179359484817, 21.80708638639346579252317416856, 22.52772678403063036240113937896, 24.23034079889668890709678651731, 24.944108972031221393528865677730, 26.54612605683037511510076788995, 27.71226770974994355972049445437, 28.49466260002041905704980662808, 30.07775608826669684024746711091, 30.94087038465126712079977342796, 31.83254574832783021442933502355

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