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Properties

Label 2-3240-360.13-c0-0-5
Degree $2$
Conductor $3240$
Sign $-0.370 + 0.929i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)5-s + (0.5 + 0.133i)7-s + (−0.707 + 0.707i)8-s + (−0.500 − 0.866i)10-s + (1.67 − 0.965i)11-s + (0.258 − 0.448i)14-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)20-s + (−0.500 − 1.86i)22-s − 1.00i·25-s + (−0.366 − 0.366i)28-s + (0.707 + 1.22i)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 + (0.5 + 0.133i)7-s + (−0.707 + 0.707i)8-s + (−0.500 − 0.866i)10-s + (1.67 − 0.965i)11-s + (0.258 − 0.448i)14-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)20-s + (−0.500 − 1.86i)22-s − 1.00i·25-s + (−0.366 − 0.366i)28-s + (0.707 + 1.22i)29-s + (−0.866 + 1.5i)31-s + (0.965 − 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-0.370 + 0.929i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ -0.370 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.700339845\)
\(L(\frac12)\) \(\approx\) \(1.700339845\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \)
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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

−8.813731767521998069212128514669, −8.374875841993555022261963025776, −6.93553957368096726080837164592, −6.11140706427783096517832429796, −5.39043950170534651249225898683, −4.70828893392109529575319335751, −3.83001846640959255791206407784, −3.00488362848851984340152916119, −1.71129099332832130864366308904, −1.16460416644912233229130136477, 1.50800330657816154851757514048, 2.67604987064960688685270688112, 3.95510401535467828059395678673, 4.36647179121525449452429474301, 5.47308205827717030307496672650, 6.17275011679719354234983655155, 6.78125502316049827651305751848, 7.38468674399473646198366808685, 8.137346885147345462425270856844, 9.147057973088083393062506384053

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