Properties

Label 1-1480-1480.403-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.555 - 0.831i$
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  + (−0.342 − 0.939i)3-s + (0.984 + 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.642 − 0.766i)17-s + (0.939 − 0.342i)19-s + (−0.173 − 0.984i)21-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)27-s + (−0.5 − 0.866i)29-s − 31-s + (−0.642 + 0.766i)33-s + (−0.939 − 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)3-s + (0.984 + 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.642 − 0.766i)17-s + (0.939 − 0.342i)19-s + (−0.173 − 0.984i)21-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)27-s + (−0.5 − 0.866i)29-s − 31-s + (−0.642 + 0.766i)33-s + (−0.939 − 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6314698873 - 1.181733760i\)
\(L(\frac12)\) \(\approx\) \(0.6314698873 - 1.181733760i\)
\(L(1)\) \(\approx\) \(0.9095122881 - 0.4623474177i\)
\(L(1)\) \(\approx\) \(0.9095122881 - 0.4623474177i\)

Euler product

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

−20.89000805510371097032610177480, −20.46391366965507062469756070057, −19.633098465616193478356408582, −18.31786704148861398430144849170, −17.990346239908981779945556519270, −17.04756426785450938685063683081, −16.46908977017052351412598998518, −15.60767474303957107250287329895, −14.90541076285797463831527636561, −14.3691399205740033635910370929, −13.399972560047950917988258849781, −12.38722223007050145025948306964, −11.594643608956030820619994770902, −10.79456171728211866799165404547, −10.44732461043099159745696850741, −9.22026031026547828077342477118, −8.80037164467841719710449906967, −7.68970871115678700767826370350, −6.86097263684893214376227469320, −5.74115530512342145216050852668, −5.02770628540151344072621109949, −4.29888729730991088480902205881, −3.574249072590415298322705419519, −2.28078680181131181771880352385, −1.27915819846028453918690718842, 0.57209112804954184440930579614, 1.48006338279332026992583171299, 2.54168013552105253878374273105, 3.35572849369914173910591969236, 4.84863113976851321710285228596, 5.4395420285519230538112526653, 6.17128715190479145881261876401, 7.28247292155128758652845189127, 7.85925214388341856956167061428, 8.55713961324373540300229655175, 9.46172850088691454485342199519, 10.93063290607530532281047127800, 11.15717926363601026802334824477, 11.84856767845405845768578775908, 13.01852714698010848563346949494, 13.40881367802437881804327612173, 14.16061937200993597247875322144, 15.10481971383456041607522221891, 15.94573586490035153998732602570, 16.74277414437032513163665663794, 17.69466367205221151240971997230, 18.14357880795144490028956317548, 18.63926154537498909171487050741, 19.63758168036980065447078611396, 20.337618780542307536054686647098

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