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Properties

Label 2-3240-360.139-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.364121090\)
\(L(\frac12)\) \(\approx\) \(1.364121090\)
\(L(1)\) 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

−8.963539446580402162451349655803, −8.434227416859890239037130989296, −7.50254475831127229309433004000, −7.06926451088116873158669743901, −6.23766759459886374181177810978, −5.44331795249980835194657454002, −4.75054168757556626589103915931, −3.97442421056181413998298643358, −2.82475044348726849358182551731, −2.18708709494009863516298752007, 0.74331751315705876175166590271, 1.47682327629579344880819374462, 2.97889075806910915176215364385, 3.71032483075024729663684762868, 4.64436685109140540531209049243, 4.99720830878694356021613685091, 5.79784469911178244281475374488, 7.18500568594211468695779536979, 7.76758315303939289276576316810, 8.328500053520705541978598917378

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