Properties

Label 2-2960-740.163-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.982 - 0.183i$
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  + (0.866 + 0.5i)5-s + (−0.642 − 0.766i)9-s + (−0.766 − 0.642i)13-s + (1.26 + 1.50i)17-s + (0.499 + 0.866i)25-s + (1.92 + 0.515i)29-s + (0.642 − 0.766i)37-s + (1.26 − 1.50i)41-s + (−0.173 − 0.984i)45-s + (−0.342 + 0.939i)49-s + (0.296 − 0.424i)53-s + (0.0151 + 0.173i)61-s + (−0.342 − 0.939i)65-s + (−1.36 + 1.36i)73-s + (−0.173 + 0.984i)81-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)5-s + (−0.642 − 0.766i)9-s + (−0.766 − 0.642i)13-s + (1.26 + 1.50i)17-s + (0.499 + 0.866i)25-s + (1.92 + 0.515i)29-s + (0.642 − 0.766i)37-s + (1.26 − 1.50i)41-s + (−0.173 − 0.984i)45-s + (−0.342 + 0.939i)49-s + (0.296 − 0.424i)53-s + (0.0151 + 0.173i)61-s + (−0.342 − 0.939i)65-s + (−1.36 + 1.36i)73-s + (−0.173 + 0.984i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.982 - 0.183i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.982 - 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421421126\)
\(L(\frac12)\) \(\approx\) \(1.421421126\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.642 + 0.766i)T \)
good3 \( 1 + (0.642 + 0.766i)T^{2} \)
7 \( 1 + (0.342 - 0.939i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
19 \( 1 + (-0.642 - 0.766i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.92 - 0.515i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.296 + 0.424i)T + (-0.342 - 0.939i)T^{2} \)
59 \( 1 + (0.342 + 0.939i)T^{2} \)
61 \( 1 + (-0.0151 - 0.173i)T + (-0.984 + 0.173i)T^{2} \)
67 \( 1 + (-0.342 + 0.939i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 + (-0.342 + 0.939i)T^{2} \)
83 \( 1 + (-0.984 - 0.173i)T^{2} \)
89 \( 1 + (0.939 + 0.657i)T + (0.342 + 0.939i)T^{2} \)
97 \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.979010416372772439434936855884, −8.242796893098948015996589739857, −7.42427101585761370232886465153, −6.56523502417828964114743109496, −5.83496874327769351457738492102, −5.42600551944500450858754232164, −4.15394570695657512492783111148, −3.16505818144193450568003275732, −2.51811238085617191230398127431, −1.18853843382571882587731570241, 1.11719719782028203895762089515, 2.43172829774818466656617828423, 2.94718704036420210746721681196, 4.59295325119970380945233329963, 4.94199575838138056118448079116, 5.79935526828650828235390577036, 6.54196576952749463480933207926, 7.51171600939045450740000132282, 8.155175924261900120968792433635, 8.956471574121515883819159663764

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