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Properties

Label 2-3240-360.139-c0-0-12
Degree $2$
Conductor $3240$
Sign $-0.984 + 0.173i$
Analytic cond. $1.61697$
Root an. cond. $1.27160$
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 + (−0.866 − 1.5i)7-s + 0.999·8-s − 0.999·10-s + (0.866 − 1.5i)13-s + (−0.866 + 1.5i)14-s + (−0.5 − 0.866i)16-s + 19-s + (0.499 + 0.866i)20-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s − 1.73·26-s + 1.73·28-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.866 − 1.5i)7-s + 0.999·8-s − 0.999·10-s + (0.866 − 1.5i)13-s + (−0.866 + 1.5i)14-s + (−0.5 − 0.866i)16-s + 19-s + (0.499 + 0.866i)20-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s − 1.73·26-s + 1.73·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-0.984 + 0.173i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (2539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ -0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8756293659\)
\(L(\frac12)\) \(\approx\) \(0.8756293659\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−8.659026593458964371784684752960, −7.72037272889714567764072734452, −7.41413401276849836555810582693, −6.14335482603793184323978007259, −5.38709226836024066927797194877, −4.31061318257815264890801525689, −3.63200469369692275170960555463, −2.90473157242538464654260305365, −1.41987828046332930239275218874, −0.67930398269369924461695605793, 1.70067361798896054887434715145, 2.62351141206643419514889320265, 3.68212175940685433749417324139, 4.86771731930080424962378256485, 5.78674862560440353804293252799, 6.36943215775817397739566990011, 6.62985748785380131110534047553, 7.67072425542563848069468683923, 8.536483799637551406100586146584, 9.300324909191601035747072368974

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