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Properties

Label 1-1480-1480.1133-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.445 - 0.895i$
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  + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)21-s − 23-s − i·27-s − i·29-s − i·31-s + (−0.866 − 0.5i)33-s + (0.866 − 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)21-s − 23-s − i·27-s − i·29-s − i·31-s + (−0.866 − 0.5i)33-s + (0.866 − 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.445 - 0.895i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.445 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9462058946 - 0.5861922811i\)
\(L(\frac12)\) \(\approx\) \(0.9462058946 - 0.5861922811i\)
\(L(1)\) \(\approx\) \(0.8681919168 - 0.1571736207i\)
\(L(1)\) \(\approx\) \(0.8681919168 - 0.1571736207i\)

Euler product

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

−20.931504530516929694846963842892, −20.033286584435161665889184470959, −19.490884367463015056498028797720, −18.22834590717909734075554188198, −17.58715330380356884644939784423, −17.189998406985718400348519176197, −16.40798843078786942941965425954, −15.57472429075989509927698945774, −14.61010575112066147569960427642, −14.3702692111173767425548158342, −12.94911064853621863223424549758, −12.34151531008797659989503989962, −11.55064190482303490947319913107, −10.690869771018431765295389911989, −10.35856596632325731853916698054, −9.29652704158481236298185668850, −8.39913918106312198537069590367, −7.5151253195395432058388186972, −6.512925846072380977421455565875, −5.87936989977165038847400300455, −4.84160434646301283744337261170, −4.24468219850772404777251345212, −3.44099262989309695004083833923, −1.89638777395573506567987561222, −1.005935546903729650533749885892, 0.55941319275484719625153865847, 1.91083516132012593238368297168, 2.303470658340772938148607093127, 4.16308146586077126920921574644, 4.5970978718681409029213376954, 5.64912122574971553751485165535, 6.38290103352370076933147203120, 7.141749445614846207888848694724, 7.96548512777495745083691470853, 8.94872679421721927114493799321, 9.69082030172493258831315520311, 10.83821027764732239824748055124, 11.58665292322563266355169884631, 11.864363574226016165762081542977, 12.75086376847266943323412359788, 13.79139455707899594842383530539, 14.32534716040882809152113419601, 15.32270248650487781742908638193, 16.08916219369545679393288831017, 17.13061915046628068152328122579, 17.34127853848054304766832735003, 18.25058790713157799912673704596, 18.9395050308703050763927287530, 19.57686979725719601329319549436, 20.58472638144025691495910397861

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