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Properties

Label 1-1480-1480.557-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.608 + 0.793i$
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  + (0.984 − 0.173i)3-s + (0.642 + 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)17-s + (0.984 − 0.173i)19-s + (0.766 + 0.642i)21-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)27-s + (−0.866 − 0.5i)29-s + i·31-s + (−0.342 + 0.939i)33-s + (−0.984 − 0.173i)39-s + (0.939 + 0.342i)41-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)3-s + (0.642 + 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)17-s + (0.984 − 0.173i)19-s + (0.766 + 0.642i)21-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)27-s + (−0.866 − 0.5i)29-s + i·31-s + (−0.342 + 0.939i)33-s + (−0.984 − 0.173i)39-s + (0.939 + 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.608 + 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.023412337 + 0.9987628412i\)
\(L(\frac12)\) \(\approx\) \(2.023412337 + 0.9987628412i\)
\(L(1)\) \(\approx\) \(1.489537603 + 0.2326093857i\)
\(L(1)\) \(\approx\) \(1.489537603 + 0.2326093857i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.642 + 0.766i)T \)
59 \( 1 + (0.642 - 0.766i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.642 + 0.766i)T \)
83 \( 1 + (-0.342 - 0.939i)T \)
89 \( 1 + (0.642 - 0.766i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−20.637655636685746869020557546396, −19.89624146417013868733242226197, −19.13207081068960746646232785779, −18.44786157993929507667206203563, −17.604086140352517169951607696952, −16.64122512514273700264406580773, −16.06267394557116752747481613040, −15.11061891454979747167101574042, −14.42534466493158717913825109680, −13.81509738712508884054807600106, −13.21840887099206674464260033989, −12.269082061184040088358721824259, −11.08109225967649186962556381778, −10.68836208935875582023916472447, −9.54477827105447551993092850602, −9.05053348295403346855253515119, −7.96034914207551968188681250269, −7.55050051905577682124600054215, −6.68003918574721317388547692226, −5.32612301133275771338653806245, −4.55873796441325985515343934990, −3.777878196650836878426993352556, −2.76672296804420018930334639316, −2.02782207046516416545332576905, −0.75475034013289162396746288836, 1.34454793013708667287881651853, 2.30512002012862203519609966683, 2.79391500001176134984307552567, 4.03889260344284962111154263193, 4.89784537210564329114414727833, 5.665663471310936026967298058835, 7.07644102961976991401680351586, 7.515349513774748854676418123400, 8.3403763864236965829474829185, 9.22391675805836811458879575343, 9.71229625062950051985476610272, 10.75874360785770602206763802096, 11.726234069328292453707259683663, 12.56486878819546568976629181200, 13.102491024938890948606228441217, 14.097461298457944143821959910177, 14.7474650170960683905093576266, 15.44606268180798476664187985124, 15.81824954405103372857416647168, 17.376711571035661111749958312537, 17.768379115003941580496439844466, 18.57000934878542626736221013161, 19.33958761369504151474490876970, 20.09268804469211910486378856798, 20.61238736262860494028541651010

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