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Properties

Label 1-1480-1480.957-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.900 - 0.434i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.900 - 0.434i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (957, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.900 - 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3617504263 - 1.581608437i\)
\(L(\frac12)\) \(\approx\) \(0.3617504263 - 1.581608437i\)
\(L(1)\) \(\approx\) \(1.001797575 - 0.6806816792i\)
\(L(1)\) \(\approx\) \(1.001797575 - 0.6806816792i\)

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

−21.02148614488056015032991138431, −20.376318962126235829802545374507, −19.50583006002437481480161682100, −18.776489740582903461392913867063, −18.19123623210122058094693784955, −17.1020007330175231664346864281, −16.16399821250453283591604738820, −15.74659744327055825609103054173, −15.05073819269445955938689393508, −14.20902805446297246015672781019, −13.60715636071014829471704542962, −12.56703927173609969324646021592, −11.8323102568709683332568398877, −10.99717466505744887818686444568, −9.907126119700960685559490480138, −9.46334393545400395686364225701, −8.83017433642089761111411483688, −7.812043285504795695803408881658, −7.0989754131148004801704772904, −5.86094786707714899746378078900, −5.09134916150755747186623724017, −4.321187068954520464003143448820, −3.26807356856046559305253677876, −2.55702007443803228852623693018, −1.65660138787649265621877527355, 0.570568799474028853864191948450, 1.3441962617847485074496540728, 2.822652715186707176868581786984, 3.1872135807723710121892359145, 4.25209329938103379964541534586, 5.456830801096731118785282361129, 6.401517858236552881027855572327, 7.04267824630328658375636693715, 8.06465458382092966503991933215, 8.37189830841900798973219474411, 9.48432856419303390481746848356, 10.4187657816610363598185247030, 11.00452298313794997516028572374, 12.16561594143392362985935019156, 12.96919783256008730535212615505, 13.41883062645268967985908291721, 14.1161487838103358434374556611, 14.95162398384873880780170927533, 15.78378946069790461090043421009, 16.60839114297859311927432826697, 17.52625166913928538260754287320, 18.081575488033845674423613444755, 18.97492693277091617184725649934, 19.668784195947476678832598614524, 20.097019208922349918421330140302

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