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Properties

Label 1-1480-1480.147-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.850 - 0.525i$
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  − i·3-s − i·7-s − 9-s + 11-s − i·13-s − i·17-s + 19-s − 21-s − i·23-s + i·27-s − 29-s + 31-s − i·33-s − 39-s + 41-s + ⋯
L(s)  = 1  − i·3-s − i·7-s − 9-s + 11-s − i·13-s − i·17-s + 19-s − 21-s − i·23-s + i·27-s − 29-s + 31-s − i·33-s − 39-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4161273951 - 1.464829610i\)
\(L(\frac12)\) \(\approx\) \(0.4161273951 - 1.464829610i\)
\(L(1)\) \(\approx\) \(0.9031452314 - 0.6412356449i\)
\(L(1)\) \(\approx\) \(0.9031452314 - 0.6412356449i\)

Euler product

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

−21.131852503979515065510136184229, −20.33401835886887004785519125881, −19.35640683987228156379279406239, −19.01038227623975324475514910674, −17.73778169040339271495618760550, −17.21602124687175638598928575632, −16.30191287214280469138216216526, −15.759139788875553924650058996417, −14.91036377925254939250153521672, −14.412947764066485693937403163994, −13.557687817315782421341557874422, −12.37630901021512045685507999458, −11.60475828581893328536427706066, −11.23780197604042629012600001932, −9.9982831440603782649271073240, −9.359383608110354963040411170054, −8.87384142982389289317130306783, −7.95772159645824757959975644563, −6.72567704700964140456543718132, −5.87442987018099598797454137905, −5.2275956935351423372934727188, −4.14314028136176539599861704874, −3.57623411938214563695067137242, −2.47580949501013067625561606782, −1.44793663606853065704975310059, 0.641540535522301554039404838474, 1.265664950819015169510394845458, 2.569505594509637274348755125759, 3.371492035733630982685071123461, 4.42985207460449764473382160949, 5.480699863912272524984003202384, 6.36229328751180785633903847193, 7.1725234005608543825851073362, 7.66289119856724833105724416385, 8.59096545611020577260402540031, 9.5226615979570096523259202653, 10.39942240919467652877705081856, 11.37859430419103030237364501832, 11.89614871479613752595773313635, 12.884301554474505245333541071589, 13.45644221085132164971819444667, 14.22973070801913416824123152045, 14.7465064955624874926274212202, 16.035112051907190658723969538982, 16.7145702376677028954230802290, 17.51061691462599265980480116819, 17.99776191662624484327956226007, 18.87406518245815372685538039150, 19.6473266470308427147394076373, 20.32591123182763270841016501368

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