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Properties

Label 2-2960-2960.877-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.716 + 0.697i$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (−0.366 − 1.36i)7-s − 0.999·8-s + 0.999·10-s + (1 − i)11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.999 − i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (−1.36 − 0.366i)21-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (−0.366 − 1.36i)7-s − 0.999·8-s + 0.999·10-s + (1 − i)11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.999 − i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (−1.36 − 0.366i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (877, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.716 + 0.697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.829544487\)
\(L(\frac12)\) \(\approx\) \(1.829544487\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-1 + i)T - iT^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T + T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.505426219567040381271505592534, −8.061689875557413560755688129974, −7.20896097423484676339364969377, −6.58652281751298125637122947873, −6.10861246519775590578475009891, −4.89232779640381181704043729427, −4.23485356235168887897103097655, −3.46526693871285102050131988672, −2.14351800040779040775867802014, −0.933770522305192184577127610965, 1.91443953037833343238994347086, 2.51822585297652514895331950761, 3.48456386823877800941004806261, 3.94068508686296449729205165612, 5.15888680910782515941135938460, 5.71495938869540365615727762790, 6.51532197080721099875161462714, 7.46266844348945936290803356838, 8.823109352774628446698623875492, 9.268180963473405842042714995055

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