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Properties

Label 2-2520-840.59-c0-0-3
Degree $2$
Conductor $2520$
Sign $0.999 + 0.0431i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.707 + 0.707i)7-s − 0.999i·8-s + (0.866 + 0.499i)10-s + (−0.448 − 0.258i)11-s + 1.93i·13-s + (0.965 + 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)19-s + 0.999·20-s − 0.517·22-s + (1.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.965 + 1.67i)26-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.707 + 0.707i)7-s − 0.999i·8-s + (0.866 + 0.499i)10-s + (−0.448 − 0.258i)11-s + 1.93i·13-s + (0.965 + 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)19-s + 0.999·20-s − 0.517·22-s + (1.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.965 + 1.67i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0431i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.999 + 0.0431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.266663512\)
\(L(\frac12)\) \(\approx\) \(2.266663512\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.93iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + 0.517T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.129027940793493974204338373633, −8.542843288339302863268420894138, −7.14325171992085428015169221546, −6.72212941656260935823103325360, −5.87397255571943141010294940807, −5.13706618662821514582783955407, −4.31749669845256259801274165789, −3.34834165223397769369811289590, −2.25472281142609356795800318986, −1.85167881225395889938509128576, 1.27668716192067675108858873325, 2.60505020667653870371918495945, 3.53699148878805670357823402289, 4.72611055418391522809757151844, 5.04887088036535691984369521087, 5.75094116908242739686370623133, 6.77829607254220110610231854790, 7.56146256162855028625925752840, 8.230392279450203419970658677830, 8.737852502264040239331051515983

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