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Properties

Label 1-2960-2960.2003-r1-0-0
Degree $1$
Conductor $2960$
Sign $0.930 - 0.366i$
Analytic cond. $318.096$
Root an. cond. $318.096$
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  + (−0.766 − 0.642i)3-s + (0.342 + 0.939i)7-s + (0.173 + 0.984i)9-s + (−0.866 + 0.5i)11-s + (−0.984 − 0.173i)13-s + (0.173 + 0.984i)17-s + (−0.766 − 0.642i)19-s + (0.342 − 0.939i)21-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)27-s + (0.5 + 0.866i)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  + (−0.766 − 0.642i)3-s + (0.342 + 0.939i)7-s + (0.173 + 0.984i)9-s + (−0.866 + 0.5i)11-s + (−0.984 − 0.173i)13-s + (0.173 + 0.984i)17-s + (−0.766 − 0.642i)19-s + (0.342 − 0.939i)21-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)27-s + (0.5 + 0.866i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.930 - 0.366i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ 0.930 - 0.366i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7618713103 - 0.1448494397i\)
\(L(\frac12)\) \(\approx\) \(0.7618713103 - 0.1448494397i\)
\(L(1)\) \(\approx\) \(0.6873978413 + 0.01616670483i\)
\(L(1)\) \(\approx\) \(0.6873978413 + 0.01616670483i\)

Euler product

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

−18.9300850244153205235911896077, −18.06210894547437208216377606393, −17.491067158539198957121332694410, −16.71490797804805338604600690560, −16.4336174283774816506035849675, −15.49507575500505027575551683321, −14.8639941283130256583920716607, −14.09072246898916885887191511612, −13.30777321622036416152644029599, −12.57398640179022565400949429017, −11.58458792039563351578932776956, −11.23676754184027635405620623944, −10.36414476374868769192109965427, −9.86781165349085876896375337934, −9.176336374916889716255668490327, −7.9012207295905277620742707911, −7.53555919741623544984112394402, −6.47899112834667061037361141959, −5.801507112769585289301352919894, −4.76878963472437746198566142493, −4.57296646743030673016328540663, −3.47231235007217565439834166992, −2.66170559322229158926508043141, −1.36157291883540043128043944716, −0.41047282824589714558788186903, 0.2975365366204975931930062096, 1.6039427813338636262731262502, 2.24880736523197765907225222115, 2.987690755175136977018141013345, 4.50011889467260896980084500412, 5.076757325780099417562857665800, 5.61683722195303681173294354459, 6.640688134793868335446393229915, 7.113129387780884261641471553752, 8.15874030418051210039407085581, 8.5524292531639011564788707102, 9.68711629170340778672177155088, 10.62091707963370634305669159364, 10.92173805684731615939193926257, 12.045136649890690280932631953917, 12.602479251176263476446816380735, 12.77356228197247971082096642235, 13.94184660116980453688456236615, 14.77251993272487254065573640723, 15.35486646101508184231530719766, 16.10989916688791485256153044636, 16.995475696777732975403241488998, 17.585387635684752878070049917162, 18.046647698784160223956547834715, 18.90601845758890941669446535153

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