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Properties

Label 2-3240-360.259-c0-0-5
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  − 2-s + 4-s + (−0.5 − 0.866i)5-s − 8-s + (0.5 + 0.866i)10-s + 16-s + 1.73i·17-s + 19-s + (−0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−1.5 + 0.866i)31-s − 32-s − 1.73i·34-s − 38-s + ⋯
L(s)  = 1  − 2-s + 4-s + (−0.5 − 0.866i)5-s − 8-s + (0.5 + 0.866i)10-s + 16-s + 1.73i·17-s + 19-s + (−0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−1.5 + 0.866i)31-s − 32-s − 1.73i·34-s − 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: $\chi_{3240} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6952086431\)
\(L(\frac12)\) \(\approx\) \(0.6952086431\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
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

−8.795877625995565746022054425171, −8.322253063691103526232715978680, −7.46180425083176276816000841178, −7.00888502137491555693564544509, −5.76722528973512968031969162656, −5.35751382373342910778605412372, −4.00060221469824650592778499589, −3.34875778136731301969430803148, −1.95272012553594927667455717285, −1.07774004749337072189208513184, 0.70540924740832342775693901353, 2.30070042155526006335616633926, 2.94601141062039916783181935240, 3.85773602517784727052389470813, 5.11847098845601104547859308480, 5.98611218435903779852507887347, 6.97886813211644625743582875593, 7.27912523533879543231904159018, 7.937226770852492521139951155570, 8.892129304618549294684432002935

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