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Properties

Label 2-3240-360.139-c0-0-4
Degree $2$
Conductor $3240$
Sign $0.939 + 0.342i$
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 + (0.499 + 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73i·17-s + 19-s + 0.999·20-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (1.5 + 0.866i)31-s + (0.499 + 0.866i)32-s + (1.49 + 0.866i)34-s + (0.5 − 0.866i)38-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 + (0.499 + 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73i·17-s + 19-s + 0.999·20-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (1.5 + 0.866i)31-s + (0.499 + 0.866i)32-s + (1.49 + 0.866i)34-s + (0.5 − 0.866i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.939 + 0.342i$
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: \((2,\ 3240,\ (\ :0),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.352561135\)
\(L(\frac12)\) \(\approx\) \(1.352561135\)
\(L(1)\) 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 \( 1 - 1.73iT - T^{2} \)
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 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 - T + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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

−8.811558973847702002655833772939, −8.186860105875462530177872551970, −7.21109525321414142132101978870, −6.37538762640902366912605595907, −5.78625466329583375377506267025, −4.68556014356963110531476053953, −3.99993929063249794312464487497, −3.17829903633742274743160476213, −2.48135939208053609214986728750, −1.21172007572193496484057924199, 0.827728302549680570199567503576, 2.65501909914820034429238370499, 3.55801591570220950716833100703, 4.44139765229329629851185928633, 5.12441084845364472506472216577, 5.63689739952118966535027718403, 6.72186047110978722716552833667, 7.50192707701956675179681701891, 7.83742917416974922249155223262, 8.872271334701190448041472904085

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