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Properties

Label 1-2960-2960.43-r1-0-0
Degree $1$
Conductor $2960$
Sign $0.241 + 0.970i$
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  + 3-s − i·7-s + 9-s + i·11-s + i·13-s + 17-s + 19-s − i·21-s − 23-s + 27-s + 29-s + i·31-s + i·33-s + i·39-s + 41-s + ⋯
L(s)  = 1  + 3-s − i·7-s + 9-s + i·11-s + i·13-s + 17-s + 19-s − i·21-s − 23-s + 27-s + 29-s + i·31-s + i·33-s + i·39-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.241 + 0.970i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ 0.241 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.772501319 + 2.168117457i\)
\(L(\frac12)\) \(\approx\) \(2.772501319 + 2.168117457i\)
\(L(1)\) \(\approx\) \(1.596255260 + 0.1985093750i\)
\(L(1)\) \(\approx\) \(1.596255260 + 0.1985093750i\)

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 \( 1 + T \)
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 \( 1 + T \)
67 \( 1 \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 \)
89 \( 1 - T \)
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

−18.638164751323320055333596132551, −18.42577640840290120252504776086, −17.53383377351466133519735671088, −16.413397585879480728428293885277, −15.85271423331323815635970732606, −15.28875012869871686564867551543, −14.5278326381012441316381020420, −13.88461869088535863296060016692, −13.30166693055193450973616261372, −12.32045845656292137116652575881, −11.95955354406041383225431240999, −10.8363297545624805385181588165, −10.04657254125042121008286498791, −9.39835589347658961510019405489, −8.666216769878377794813505766408, −7.959463325292687687275673459788, −7.584528520366231638211096321905, −6.24916216187717892717326771077, −5.69487565809282863484202296390, −4.85422771574058556453261550872, −3.67387731376375070001934086598, −3.10915074809537705801991683502, −2.47552207616684145329341034477, −1.44170592336898417119602114141, −0.46427653720628889782787971860, 1.139758382160469935026435570079, 1.61895721098169493304105923873, 2.7470698389886916135684717504, 3.47165626136540962745511071999, 4.36381990160657516197945868740, 4.755694727135289997557202684, 6.14287066580374793640216581035, 7.001292282131891894654458384728, 7.56675054906638578383118598658, 8.10226877099066779240519519431, 9.18507176245267853483388961888, 9.77804111853874760026591644773, 10.22331452491731648112859897305, 11.21586544445106066475218274133, 12.23177109323209962983415798906, 12.66897184172043048572797186360, 13.74967059738418975728128641073, 14.18733952164376727716121439686, 14.50877510012177370732361291558, 15.70285379380494762832757618871, 16.09134797576948908600666534867, 16.92214090840687206159062026923, 17.81326129134223527533755535685, 18.35859796821945291209674820385, 19.27987517987389248153838505323

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