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Properties

Label 1-1480-1480.1469-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.989 + 0.146i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)21-s − 23-s + 27-s − 29-s + 31-s + (0.5 + 0.866i)33-s + (−0.5 + 0.866i)39-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)21-s − 23-s + 27-s − 29-s + 31-s + (0.5 + 0.866i)33-s + (−0.5 + 0.866i)39-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.989 + 0.146i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.989 + 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01568221815 - 0.2128307378i\)
\(L(\frac12)\) \(\approx\) \(0.01568221815 - 0.2128307378i\)
\(L(1)\) \(\approx\) \(0.6944007980 - 0.1649494522i\)
\(L(1)\) \(\approx\) \(0.6944007980 - 0.1649494522i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−21.11049619893480765038395749942, −20.47411613048498445270851691274, −19.71447640531744152189190980985, −18.77568623459993283384087160070, −17.73997116734827507350913198584, −17.35424291559383956656780404392, −16.44935159012465869139823991518, −15.97563555136990383514480465913, −14.998538150484987911594353296015, −14.4018085945410598690156608020, −13.544954398373188855980504449851, −12.65120683524933462446802305403, −11.56695709156216126671819487631, −11.19571880527002322568726520726, −10.082650222892828228258525917, −9.931210357600576749957765337498, −8.69914306057055087545872741353, −7.850874218386591581316568683406, −6.99687270919111146573122395479, −6.00254935723195289793200965126, −5.1135635980151494322208381268, −4.44665208100295110382532443073, −3.70930761751718343635122383465, −2.60384722118546512246338213101, −1.3313691952670408151916761523, 0.08661121336054289709167039628, 1.445553195537027892382377211340, 2.38681478199383243037582026125, 3.084531128333389644863581733642, 4.65575645993154591568612733840, 5.61927033474742861433503594315, 5.71091062764002344508347935505, 7.10085608996737396113032970931, 7.88961824676115342666958070376, 8.21953262699889071388104162460, 9.52959782366720566773375829024, 10.35988965493714589834121427476, 11.21439482556973011312805900516, 12.11971694694043093625825331473, 12.37732076936191093058883684409, 13.38343077647096708347628151730, 14.09571107343558561363945628339, 14.964998076515176505616279428739, 15.84714360746877992193956133204, 16.493552356580290894970153138626, 17.61489544342606196108661099284, 17.975792872497148180874326337562, 18.69478658858033488634593076876, 19.21848478895536605226692742475, 20.43081518602962842274451078572

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