Properties

Label 2-2520-280.219-c0-0-4
Degree $2$
Conductor $2520$
Sign $0.553 + 0.832i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s i·7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (−0.866 − 1.5i)11-s + 1.73·13-s + (0.866 + 0.5i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·20-s + 1.73·22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.866 + 1.49i)26-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s i·7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (−0.866 − 1.5i)11-s + 1.73·13-s + (0.866 + 0.5i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·20-s + 1.73·22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.866 + 1.49i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (2179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8987987165\)
\(L(\frac12)\) \(\approx\) \(0.8987987165\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + iT \)
good11 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.73T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.73T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−8.708202029244999228751774035617, −8.275371553223509426037140250954, −7.69094342734991405660222969710, −6.57353155962901573915586939451, −5.79523984363789229939104948339, −5.50491354071707484736288524031, −4.24537433939005443392608169263, −3.56123711740943228490597015783, −1.70773828801080606897512908778, −0.73153284600720958533665269965, 1.71264965498424671970413934753, 2.44368414825167346409632343769, 3.16709552451382687345996459902, 4.31932665036546984463636217158, 5.18752993853505047890348709297, 6.28154495251464900403373981457, 6.90793447791917104662915218013, 7.975990845762514812652408472729, 8.549790136339302155564244480718, 9.332408748258490948486748361630

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