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Properties

Label 1-2960-2960.69-r1-0-0
Degree $1$
Conductor $2960$
Sign $0.930 + 0.367i$
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  + (0.642 − 0.766i)3-s + (−0.939 + 0.342i)7-s + (−0.173 − 0.984i)9-s + (−0.866 + 0.5i)11-s + (0.173 − 0.984i)13-s + (−0.984 + 0.173i)17-s + (−0.766 − 0.642i)19-s + (−0.342 + 0.939i)21-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)27-s + (−0.5 − 0.866i)29-s + i·31-s + (−0.173 + 0.984i)33-s + (−0.642 − 0.766i)39-s + (0.173 − 0.984i)41-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)3-s + (−0.939 + 0.342i)7-s + (−0.173 − 0.984i)9-s + (−0.866 + 0.5i)11-s + (0.173 − 0.984i)13-s + (−0.984 + 0.173i)17-s + (−0.766 − 0.642i)19-s + (−0.342 + 0.939i)21-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)27-s + (−0.5 − 0.866i)29-s + i·31-s + (−0.173 + 0.984i)33-s + (−0.642 − 0.766i)39-s + (0.173 − 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.930 + 0.367i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ 0.930 + 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7626716077 + 0.1451359677i\)
\(L(\frac12)\) \(\approx\) \(0.7626716077 + 0.1451359677i\)
\(L(1)\) \(\approx\) \(0.8595680156 - 0.2664807093i\)
\(L(1)\) \(\approx\) \(0.8595680156 - 0.2664807093i\)

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 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 - T \)
79 \( 1 + (0.342 + 0.939i)T \)
83 \( 1 + (-0.984 + 0.173i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−18.871392665784400112200474914018, −18.53765487636859196400036422810, −17.23685743490255423610831557757, −16.46306041284286564550664513317, −16.24160030329765506952432395381, −15.38276684030931463660388786491, −14.73871461543110815193803026010, −13.973045337689193314467489847674, −13.14799183277234671342135755871, −12.95288683537297203081637544164, −11.594356887987480210714783510507, −10.85088681988190109313929834379, −10.353852318538295339293861088578, −9.48423076082799914695062995905, −8.94740859600785406111975377321, −8.25480734308309604577535161634, −7.34148415350210155553852110534, −6.53546632393479778879159352279, −5.74410099040908266284077656916, −4.666459000198225313765045624977, −4.15125875910246536403563619402, −3.23273660991391332420974325250, −2.63910861855266755770870884547, −1.67364771572508095709743070958, −0.17709892242959436333273189646, 0.519202555563921181764439184827, 1.77621501944899525047945169513, 2.588470617216051041219811264173, 3.09978655275002009648781943176, 4.044118814969572379686404516867, 5.163424059402809822388719416714, 5.962517756329836928258554795937, 6.793586284217906378696644110904, 7.29581300348424356295319204963, 8.23305033389801369232041795745, 8.82664751838586051291956804884, 9.52845237179426147544876158722, 10.37684721600230072699308834406, 11.114829516240332115291784780490, 12.20062707764051062922261978227, 12.78934450676256677039901085566, 13.22771341832418682790569589696, 13.75546338992497641498018797244, 15.02243306020690813493061827985, 15.33010490256616325978630840008, 15.859961471826982175279527125005, 17.15668024211147028565481799293, 17.59563393774868633038410293782, 18.42034029850409648135363734633, 18.934154929397540802883786571219

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