Properties

Label 2-2960-740.467-c0-0-0
Degree $2$
Conductor $2960$
Sign $-0.501 + 0.864i$
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  + i·5-s + (−0.866 − 0.5i)9-s + (−1 − 1.73i)13-s + (−1.5 − 0.866i)17-s − 25-s + (−0.366 − 0.366i)29-s + (0.866 − 0.5i)37-s + (−1.5 + 0.866i)41-s + (0.5 − 0.866i)45-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)53-s + (1.86 + 0.5i)61-s + (1.73 − i)65-s + (1 + i)73-s + (0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + i·5-s + (−0.866 − 0.5i)9-s + (−1 − 1.73i)13-s + (−1.5 − 0.866i)17-s − 25-s + (−0.366 − 0.366i)29-s + (0.866 − 0.5i)37-s + (−1.5 + 0.866i)41-s + (0.5 − 0.866i)45-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)53-s + (1.86 + 0.5i)61-s + (1.73 − i)65-s + (1 + i)73-s + (0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.501 + 0.864i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ -0.501 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4548068330\)
\(L(\frac12)\) \(\approx\) \(0.4548068330\)
\(L(1)\) 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 \( 1 - iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
31 \( 1 + iT^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + (-0.866 - 0.5i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−8.534212107925110184689276793737, −7.961164256469178390680005997379, −7.06689421597939313226189128738, −6.52551735859523822814952566151, −5.61661105426379618982760777954, −4.94193320433058939994816512731, −3.71534200538352410732018693719, −2.88273374432666067305233115769, −2.33442474896933231737728510500, −0.25572418747778225709090047297, 1.72991289785876279656360418573, 2.40232501132904778095962122592, 3.84076512655674532085631246564, 4.62427354638850148906312380742, 5.14171191864966072784230624461, 6.18588692174135551500947673952, 6.84504634075586837777814614243, 7.82647992333480412773770700716, 8.558803216402892824481653024456, 9.073316607538249896801854410719

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