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Properties

Label 1-1480-1480.59-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: $\chi_{1480} (59, \cdot )$
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

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

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