↑

Properties

Label 1-1480-1480.507-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.918 + 0.395i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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)3-s + (−0.866 + 0.5i)7-s + (0.5 − 0.866i)9-s + 11-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)21-s − i·23-s + i·27-s + 29-s − 31-s + (−0.866 + 0.5i)33-s + (−0.5 + 0.866i)39-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.866 + 0.5i)7-s + (0.5 − 0.866i)9-s + 11-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)21-s − i·23-s + i·27-s + 29-s − 31-s + (−0.866 + 0.5i)33-s + (−0.5 + 0.866i)39-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.918 + 0.395i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.918 + 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031261105 + 0.2125236728i\)
\(L(\frac12)\) \(\approx\) \(1.031261105 + 0.2125236728i\)
\(L(1)\) \(\approx\) \(0.8120581955 + 0.1285576488i\)
\(L(1)\) \(\approx\) \(0.8120581955 + 0.1285576488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 + T \)
31 \( 1 - T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + iT \)
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

−20.44002542876167216378752147858, −19.71371161503574728367347738188, −19.15609198031006609316543061751, −18.27779314277756125739052341943, −17.60539552224366338587547053021, −16.84300998835423306155063615512, −16.245383459298988646723752225320, −15.62552001794742187900509524005, −14.38713770282840725887628172095, −13.62299677629702016472754932689, −12.95579422435229710795873370036, −12.30457581033756476503162380152, −11.275584047973134988223220880905, −10.93117533182295784463146024608, −9.87249218614622966619468719777, −9.0745218254960585271725935965, −8.14212292722580111872407801828, −6.882175293712051435947424291, −6.63415872909330877050055824942, −5.93434412292809960679328290462, −4.659356047995538802032975503624, −4.04202451163798731424490802094, −2.88547081949781236024536210635, −1.63452935199848842335803946598, −0.75450080980258456930415400403, 0.70737273374563852512851100447, 1.915099694796826047168180314904, 3.46155376747728848243278859997, 3.74501473264900013717221078344, 5.04993970955494483199781922542, 5.76049606707156395351711564518, 6.471862807340297778999937144206, 7.14978514601475169568052043790, 8.54888434240294969223219101199, 9.2539647143874032611566999527, 9.92599108902021011791366528708, 10.75367235374198006247084452539, 11.63122324177131854793581396139, 12.12314534184508485238398294636, 13.00565758522190461399866130051, 13.8238823622803368028469296631, 14.86735650696825232623105375185, 15.76062358291886656640917187169, 16.01598753546034020669799879572, 16.92191580178823286374133861952, 17.68105161518033368117049240963, 18.345896963925319108356940341883, 19.13139369204089921374784314976, 20.0803472141893353310468690125, 20.68504716279966308141478842723

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