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Properties

Label 2-2520-280.277-c0-0-0
Degree $2$
Conductor $2520$
Sign $0.999 + 0.0333i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.965 + 0.258i)5-s + (0.5 − 0.866i)7-s + (−0.707 − 0.707i)8-s + 10-s + (1.67 + 0.965i)11-s + (−0.707 + 0.707i)14-s + (0.500 + 0.866i)16-s + (−0.965 − 0.258i)20-s + (−1.36 − 1.36i)22-s + (0.866 − 0.499i)25-s + (0.866 − 0.5i)28-s − 0.517·29-s + (−0.866 + 1.5i)31-s + (−0.258 − 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.965 + 0.258i)5-s + (0.5 − 0.866i)7-s + (−0.707 − 0.707i)8-s + 10-s + (1.67 + 0.965i)11-s + (−0.707 + 0.707i)14-s + (0.500 + 0.866i)16-s + (−0.965 − 0.258i)20-s + (−1.36 − 1.36i)22-s + (0.866 − 0.499i)25-s + (0.866 − 0.5i)28-s − 0.517·29-s + (−0.866 + 1.5i)31-s + (−0.258 − 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0333i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.999 + 0.0333i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7587132239\)
\(L(\frac12)\) \(\approx\) \(0.7587132239\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + 0.517T + T^{2} \)
31 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
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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

−9.058906509490772761742964961264, −8.363056613907227356056253422815, −7.56631581500139933456494292205, −6.96525464172557248735218359019, −6.58209595551586921755440567428, −5.00888891072148692714153064603, −3.90488932883484361565214732586, −3.60099376210571261368213442612, −2.06593724486389304734885033263, −1.05213277002621685214835069216, 0.929337374003920926804880342409, 2.10873705728241442175053334312, 3.36789450130609570608527097869, 4.24438025809869032991773447252, 5.48127362863438466121608742871, 6.07875398093213215944003548417, 6.99134254474533736905494799464, 7.74444970425880329753526611049, 8.453865775706735788475615539451, 9.018625716733071880784722209057

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