↑

Properties

Label 1-42e2-1764.319-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.970 - 0.239i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)11-s + (0.0747 + 0.997i)13-s + (−0.988 − 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.365 + 0.930i)29-s + (0.5 − 0.866i)31-s + (0.365 + 0.930i)37-s + (−0.733 + 0.680i)41-s + (0.733 + 0.680i)43-s + (−0.0747 − 0.997i)47-s + (0.365 − 0.930i)53-s + (0.222 − 0.974i)55-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)11-s + (0.0747 + 0.997i)13-s + (−0.988 − 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.365 + 0.930i)29-s + (0.5 − 0.866i)31-s + (0.365 + 0.930i)37-s + (−0.733 + 0.680i)41-s + (0.733 + 0.680i)43-s + (−0.0747 − 0.997i)47-s + (0.365 − 0.930i)53-s + (0.222 − 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.970 - 0.239i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.988017463 - 0.2419417731i\)
\(L(\frac12)\) \(\approx\) \(1.988017463 - 0.2419417731i\)
\(L(1)\) \(\approx\) \(1.063881774 - 0.1192685185i\)
\(L(1)\) \(\approx\) \(1.063881774 - 0.1192685185i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.733 + 0.680i)T \)
43 \( 1 + (0.733 + 0.680i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (0.733 + 0.680i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.0747 + 0.997i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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

−19.937063194518063405219368435377, −19.35745501061320287580148897447, −18.698082224422073131185532699023, −17.652237447488354116007623375516, −17.53406821108121316147992109136, −16.229359675288330021171100360033, −15.59960174105218422947089755746, −14.955214328740398746501702013471, −14.038808761713167089524211211379, −13.68016523897683471578991794630, −12.47817905019812559971577976886, −11.79754199588830897096877955627, −11.0388063593659335124760573799, −10.37180216116179674012231001947, −9.60165306835679882802734798263, −8.61447558240139963076121416843, −7.82114415976671292920440799890, −7.07561389048239841920342475196, −6.1715971374853404196286056737, −5.64459816974680715150138082354, −4.26971232814595427152831833052, −3.59826413236055681595518147641, −2.79888165560318845109858473852, −1.766994568381431863193905297996, −0.56778936932149181016583737734, 0.628664374286249553078221498, 1.556770786528282587449960960110, 2.46785131105053553498743278630, 3.79871202590713157532437270281, 4.5017027094008896125845042865, 5.03511492945082750997337013317, 6.36344615602625798010959973929, 6.831298869822043176384399770923, 7.932812166157620282864574824826, 8.78683118077303986412540207610, 9.26415943251203331299839756247, 10.047544674465784053977176331636, 11.32632540946601481959603541828, 11.72608297704571803358870092535, 12.50733008595664089620985282048, 13.3602416395465779994927637562, 13.96001004640391784796405781160, 14.92532781992094368596048814587, 15.64516136877091461675517546674, 16.49020160472907540702976294235, 16.90137237201837235209402708634, 17.808320207483435753254978038285, 18.4825509241433750269865506954, 19.68011791256994103917282351194, 19.85173656743710222491035965876

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