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Properties

Label 2-3240-360.139-c0-0-1
Degree $2$
Conductor $3240$
Sign $-0.766 + 0.642i$
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 − 0.999·8-s − 0.999·10-s + (−0.5 − 0.866i)16-s − 2·19-s + (−0.499 − 0.866i)20-s + (−1 + 1.73i)23-s + (−0.499 − 0.866i)25-s + (0.499 − 0.866i)32-s + (−1 − 1.73i)38-s + (0.499 − 0.866i)40-s − 1.99·46-s + (1 + 1.73i)47-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·8-s − 0.999·10-s + (−0.5 − 0.866i)16-s − 2·19-s + (−0.499 − 0.866i)20-s + (−1 + 1.73i)23-s + (−0.499 − 0.866i)25-s + (0.499 − 0.866i)32-s + (−1 − 1.73i)38-s + (0.499 − 0.866i)40-s − 1.99·46-s + (1 + 1.73i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-0.766 + 0.642i$
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: \((2,\ 3240,\ (\ :0),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6916054690\)
\(L(\frac12)\) \(\approx\) \(0.6916054690\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 2T + T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + 2T + T^{2} \)
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 \( 1 - T^{2} \)
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

−9.107945806070341596205384419337, −8.220191972309204464731607594332, −7.74061861015608983357342069506, −7.00879746226079592237490386095, −6.27331249107494118409786574110, −5.75933731004623511008776990106, −4.58708538115479344611249199938, −3.95916538278348311561403832958, −3.19572014490340783972202313689, −2.13718998582099974750827941469, 0.33137805378116044613693867299, 1.75585252967178759348593016876, 2.60480943120325669523030033496, 3.87744489014455834965508591658, 4.32398467628636976101486490471, 5.02751787678597378330976269021, 6.01154259685017109016185108092, 6.62265656810374303411003644666, 7.891194928326787477040622995631, 8.609542266984404350782876526490

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