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Properties

Label 2-2960-740.267-c0-0-0
Degree $2$
Conductor $2960$
Sign $-0.225 - 0.974i$
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 + (1.36 + 1.36i)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 + (1.36 + 0.366i)53-s + (0.133 + 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 + (1.36 + 1.36i)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 + (1.36 + 0.366i)53-s + (0.133 + 0.5i)61-s + (−1.73 − i)65-s + (1 + i)73-s + (0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.225 - 0.974i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ -0.225 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.026861346\)
\(L(\frac12)\) \(\approx\) \(1.026861346\)
\(L(1)\) 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 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
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 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.076272855334470628651747580300, −8.576594518919719397784705823957, −7.28668978798839219591266298498, −6.77164544150487232666072710028, −6.58886104445699919501161493493, −5.24204679272870194203081855563, −4.27133937006679934405473060830, −3.75628766356101779474482366043, −2.47564802447073219848628935294, −1.74630964731061143973712143792, 0.61885483221199919356843300713, 2.01028630143822912590731975715, 2.92393240870192773345033981885, 4.25131966964184405040302770179, 4.85727840327429104113075498250, 5.37565894063459090078569825837, 6.50903850042943564346796176245, 7.31091039699011695690986881795, 8.051929221795268452538962776366, 8.571294475294532371049750829104

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