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Properties

Label 2-2960-740.183-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.221 + 0.975i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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  + (−0.939 − 0.342i)5-s + (0.984 − 0.173i)9-s + (−0.984 − 0.173i)13-s + (−0.118 − 0.673i)17-s + (0.766 + 0.642i)25-s + (0.424 − 1.58i)29-s + (0.173 − 0.984i)37-s + (0.673 + 0.118i)41-s + (−0.984 − 0.173i)45-s + (−0.642 − 0.766i)49-s + (−0.816 − 1.75i)53-s + (0.657 + 0.939i)61-s + (0.866 + 0.5i)65-s + (−0.366 + 0.366i)73-s + (0.939 − 0.342i)81-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s + (0.984 − 0.173i)9-s + (−0.984 − 0.173i)13-s + (−0.118 − 0.673i)17-s + (0.766 + 0.642i)25-s + (0.424 − 1.58i)29-s + (0.173 − 0.984i)37-s + (0.673 + 0.118i)41-s + (−0.984 − 0.173i)45-s + (−0.642 − 0.766i)49-s + (−0.816 − 1.75i)53-s + (0.657 + 0.939i)61-s + (0.866 + 0.5i)65-s + (−0.366 + 0.366i)73-s + (0.939 − 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.221 + 0.975i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1663, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.221 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9258198268\)
\(L(\frac12)\) \(\approx\) \(0.9258198268\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-0.173 + 0.984i)T \)
good3 \( 1 + (-0.984 + 0.173i)T^{2} \)
7 \( 1 + (0.642 + 0.766i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.984 + 0.173i)T + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.118 + 0.673i)T + (-0.939 + 0.342i)T^{2} \)
19 \( 1 + (-0.984 + 0.173i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.424 + 1.58i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + iT^{2} \)
41 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.816 + 1.75i)T + (-0.642 + 0.766i)T^{2} \)
59 \( 1 + (-0.642 + 0.766i)T^{2} \)
61 \( 1 + (-0.657 - 0.939i)T + (-0.342 + 0.939i)T^{2} \)
67 \( 1 + (-0.642 - 0.766i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
79 \( 1 + (0.642 + 0.766i)T^{2} \)
83 \( 1 + (0.342 + 0.939i)T^{2} \)
89 \( 1 + (0.766 + 1.64i)T + (-0.642 + 0.766i)T^{2} \)
97 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.685805475425224299087779086283, −7.914684999381379906966077240322, −7.32414197053144543866591848569, −6.74081080880867129415060190083, −5.60201082176862257549836271687, −4.66425379466120459046604829643, −4.21518284449474808133160020368, −3.20230434421922222590642461381, −2.09242843542052200090891161987, −0.62064369845028525906535050371, 1.37612287908296456623089944086, 2.64050120919898978378441172746, 3.56934701159955349369367814544, 4.46012713656033785446862537266, 4.96187238782297052887216446118, 6.25353405651951843824896443644, 6.96520267335985414676294431601, 7.55235170886426318764225173645, 8.179183149673260226034137980337, 9.076292425774759946319759448308

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