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Properties

Label 1-1480-1480.1059-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.849 - 0.527i$
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  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)21-s − i·23-s + 27-s − i·29-s − i·31-s + (0.5 − 0.866i)33-s + (0.866 − 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)21-s − i·23-s + 27-s − i·29-s − i·31-s + (0.5 − 0.866i)33-s + (0.866 − 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.849 - 0.527i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.849 - 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3855803695 - 0.1099828306i\)
\(L(\frac12)\) \(\approx\) \(0.3855803695 - 0.1099828306i\)
\(L(1)\) \(\approx\) \(0.5939327497 + 0.1921892885i\)
\(L(1)\) \(\approx\) \(0.5939327497 + 0.1921892885i\)

Euler product

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

−20.61096113388037522712142407787, −19.852648967250699591659534735, −19.14659234578167846826262822591, −18.52797852855354588559820696188, −17.73631390947914323312797219565, −16.90296280060955598211381789296, −16.53559742667364345485295574527, −15.54499018844179484014998294292, −14.52693515763480253842406232853, −13.6924604232273845988494156469, −13.08296249985344483354986659679, −12.50774470133811461216214177418, −11.57846634796909388646806887019, −10.77830470107614988062237587401, −10.15067241659868414323038508561, −9.151759223061945563686734361938, −7.95332884221418745483972409489, −7.514889512255274055435601452226, −6.5721042468104735358387953755, −6.05476846570183248142221621819, −4.8181271096593746099332653958, −4.2283160128352187628308900321, −2.696668471702350485525252633780, −2.17123744507742114042659012952, −0.747732679430399409879929557439, 0.22003203272803682752313057993, 2.10873638807366402671713618318, 2.931774335543905711024748564044, 3.834495628938095273045881052928, 5.01520146953331800143635437013, 5.36199726646740404677214438322, 6.32837449451264779788916109580, 7.1985447901931358182729100030, 8.44054563375828708170577649003, 9.04468630926865737595958514825, 9.887752268364864593372980780374, 10.62799791176041756658814892377, 11.2103153362051251522801750103, 12.42231854687999273701169741845, 12.62793893705029614924646088134, 13.79154277571384599030094781009, 14.92866906279857131404240597251, 15.43073678606550815601254956954, 15.86312383602141372512362555179, 16.82074472246602840784764421771, 17.62538146846000089705903151133, 18.09994397119574901384513445549, 19.24302515472753756992374474204, 19.79713606832715536778370982459, 20.7739943174966125464393882649

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