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Properties

Label 2-2960-740.607-c0-0-0
Degree $2$
Conductor $2960$
Sign $-0.407 - 0.913i$
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  + (0.173 + 0.984i)5-s + (0.642 + 0.766i)9-s + (−0.642 + 0.766i)13-s + (−1.50 + 1.26i)17-s + (−0.939 + 0.342i)25-s + (−1.92 − 0.515i)29-s + (0.766 + 0.642i)37-s + (1.26 − 1.50i)41-s + (−0.642 + 0.766i)45-s + (0.342 − 0.939i)49-s + (−0.424 − 0.296i)53-s + (0.0151 + 0.173i)61-s + (−0.866 − 0.5i)65-s + (1.36 + 1.36i)73-s + (−0.173 + 0.984i)81-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)5-s + (0.642 + 0.766i)9-s + (−0.642 + 0.766i)13-s + (−1.50 + 1.26i)17-s + (−0.939 + 0.342i)25-s + (−1.92 − 0.515i)29-s + (0.766 + 0.642i)37-s + (1.26 − 1.50i)41-s + (−0.642 + 0.766i)45-s + (0.342 − 0.939i)49-s + (−0.424 − 0.296i)53-s + (0.0151 + 0.173i)61-s + (−0.866 − 0.5i)65-s + (1.36 + 1.36i)73-s + (−0.173 + 0.984i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.407 - 0.913i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ -0.407 - 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031216681\)
\(L(\frac12)\) \(\approx\) \(1.031216681\)
\(L(1)\) 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 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.766 - 0.642i)T \)
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 \( 1 + (0.642 - 0.766i)T + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (1.50 - 1.26i)T + (0.173 - 0.984i)T^{2} \)
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 \( 1 + (0.424 + 0.296i)T + (0.342 + 0.939i)T^{2} \)
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 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.317127474386364637456550628186, −8.331831740528093746946175709429, −7.51415076440774636314808767206, −6.95794551160922864817177149673, −6.26363542764500780275852919246, −5.39211029247348482429253582433, −4.30465596518625551457533739545, −3.79079890070441857327940553379, −2.31127078590189232355145055465, −1.97854019399368039297541437615, 0.60921377676971690147448203303, 1.90079333116494164479640984549, 2.98043453454490261482653429224, 4.17225224174405164669164270188, 4.71774284562705136574948921367, 5.57405798591545291526322954765, 6.37545969646562427473061966591, 7.32724987701664397994767567133, 7.82166779703515914078069837965, 8.965930183497481702578305819841

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