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Properties

Label 1-1480-1480.1317-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.791 - 0.611i$
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  + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.173 + 0.984i)17-s + (0.642 + 0.766i)19-s + (−0.939 + 0.342i)21-s + (−0.5 − 0.866i)23-s + (0.866 − 0.5i)27-s + (0.866 + 0.5i)29-s − i·31-s + (0.984 − 0.173i)33-s + (−0.642 + 0.766i)39-s + (−0.173 − 0.984i)41-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.173 + 0.984i)17-s + (0.642 + 0.766i)19-s + (−0.939 + 0.342i)21-s + (−0.5 − 0.866i)23-s + (0.866 − 0.5i)27-s + (0.866 + 0.5i)29-s − i·31-s + (0.984 − 0.173i)33-s + (−0.642 + 0.766i)39-s + (−0.173 − 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.791 - 0.611i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.791 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2605981846 - 0.7640064149i\)
\(L(\frac12)\) \(\approx\) \(0.2605981846 - 0.7640064149i\)
\(L(1)\) \(\approx\) \(0.7324784566 - 0.3109102782i\)
\(L(1)\) \(\approx\) \(0.7324784566 - 0.3109102782i\)

Euler product

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

−21.291544589966472866563838030605, −20.349400058787947743074509418689, −19.45258808518652921929320374670, −18.48213394429483736033796562900, −18.01375805886960332460822596040, −17.193546379047718549955042137285, −16.19894299876552988007941258443, −15.85905870584180561233861940026, −15.15340831061933018362163807791, −14.13113816406194601337638489448, −13.54477828042083483374859015801, −12.226957909042919568020807862253, −11.67274489221143007406310189790, −11.20026803911305979532645799501, −10.21475925817103163981707027797, −9.290657444755641831989891049089, −8.88477817147926245710893426698, −7.78242532583048475902206827871, −6.67861042643293376615530958635, −5.89188130408238634012914312539, −5.0861876918247532298200449384, −4.55888078720883977537865300600, −3.29722473270864219936962483284, −2.57017811769079634650929898665, −1.16422612321793977682517170040, 0.36400014257154069865533646958, 1.48929327073282809542655169440, 2.31150518194437169835646733782, 3.59996860535513503677872648233, 4.635248512974020913405285365349, 5.35863833673211864807881102672, 6.29312999304969321828963840779, 7.10363156259968207836489171714, 7.868978415305492079704009589112, 8.30952368639416035354275278259, 9.94891263542585708408096685088, 10.40958549680770678926862426149, 11.08730454143531638120882710151, 12.21120196440897750188469694313, 12.60973042518073524208310342632, 13.440715288261034352394184411244, 14.1705248217035205854314129750, 15.05098401058919029364947790466, 15.95150905879183198066391933205, 16.91410551968916526814226713565, 17.32817733378118768816374016776, 18.11940569555501942781282100930, 18.6111197532211153130623476192, 19.79703667398147717635218720947, 20.17057830525157818180928585174

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