Properties

Label 2-2520-280.37-c0-0-0
Degree $2$
Conductor $2520$
Sign $-0.815 + 0.578i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 + 0.707i)5-s + (−0.866 + 0.5i)7-s + (−0.707 − 0.707i)8-s + (−0.500 + 0.866i)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 + 1.00i·25-s + (0.5 − 0.866i)28-s − 0.517·29-s + (−0.866 − 1.5i)31-s + (0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 + 0.707i)5-s + (−0.866 + 0.5i)7-s + (−0.707 − 0.707i)8-s + (−0.500 + 0.866i)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 + 1.00i·25-s + (0.5 − 0.866i)28-s − 0.517·29-s + (−0.866 − 1.5i)31-s + (0.965 + 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.815 + 0.578i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.815 + 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7000079715\)
\(L(\frac12)\) \(\approx\) \(0.7000079715\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
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 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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

−9.478409459808919793313929671428, −8.825474370637766754151540962456, −7.58312729728341608285350704460, −7.40455968619965672763164207135, −6.37303479969955658194174475402, −5.75178724416478003093335808319, −5.18389816726168536198968516451, −4.08059493236769137711491704605, −2.97314405148527834466448787945, −2.29361461478496766704498569028, 0.39455718704857161821142059689, 1.78438979412067062608570529992, 2.87849060130614099263787233113, 3.53140805403754522149035967925, 4.69407011316291590601422965014, 5.42447571620648521856351225043, 5.94527449868047832204880892790, 7.06157544868065204109860235348, 8.247655059362661764922212395160, 8.722162862412152166667729996201

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