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Properties

Label 1-1480-1480.1179-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.994 + 0.103i$
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  + (0.766 + 0.642i)3-s + (−0.939 + 0.342i)7-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)19-s + (−0.939 − 0.342i)21-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (−0.173 + 0.984i)33-s + (−0.642 − 0.766i)39-s + (−0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (−0.939 + 0.342i)7-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)19-s + (−0.939 − 0.342i)21-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (−0.173 + 0.984i)33-s + (−0.642 − 0.766i)39-s + (−0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.994 + 0.103i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.994 + 0.103i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04852511950 + 0.9335598269i\)
\(L(\frac12)\) \(\approx\) \(0.04852511950 + 0.9335598269i\)
\(L(1)\) \(\approx\) \(0.9205732382 + 0.4437150447i\)
\(L(1)\) \(\approx\) \(0.9205732382 + 0.4437150447i\)

Euler product

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

−20.15166214667910106344095328661, −19.46282857121086766105686817205, −19.00378781934914614864111666370, −18.27149529577529650856406403064, −17.216190802455083811909585771415, −16.64322657379197457931765778733, −15.69930952729202944916476075231, −14.93139213790552346873956839858, −13.98921207389252279779274402364, −13.65303918156446729816080902740, −12.68410048804236164722246340287, −12.16335276872742538170404798346, −11.15320807604878111675023163309, −10.13280374969131698995844938232, −9.2890611084295939473132700090, −8.800932898717844988297539938968, −7.75773258317009012349405049463, −6.94743652981451738956822950805, −6.46802418245671613960690072218, −5.37196632883659082591497185703, −4.08457599890407846564015794675, −3.3424398375696086778527686608, −2.58701968804656227311572116708, −1.514029777899320661218331802689, −0.293830064334812188906965901004, 1.679296383834174399914713574498, 2.68557318083977921577660006135, 3.257945663586868273066808641574, 4.4090783369972791386179229719, 4.94785578102865279005264318250, 6.147776511123227953542280138344, 7.13577324818644507824648837893, 7.726225041232045225789921411404, 9.07937497633324247999746211739, 9.3245580613339243096976888281, 9.98270292963319684084622151334, 10.993904787869632140652166826677, 11.84744550611591805909230220259, 13.00405541790262528392692279495, 13.217209453271007355948238232675, 14.45199852046785501204330293938, 15.070504984971363862066561891932, 15.54321584800096468437935268913, 16.432257196164052684309100532, 17.15656486455556894436885175601, 18.02759268133920769364754646744, 19.05141217830213470928663128252, 19.76722938174489484374326344377, 20.014763271271337341362457751701, 20.97255318843920281622033153591

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