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Properties

Label 2-3240-9.7-c1-0-3
Degree $2$
Conductor $3240$
Sign $-0.766 - 0.642i$
Analytic cond. $25.8715$
Root an. cond. $5.08640$
Motivic weight $1$
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)5-s + (−2 + 3.46i)11-s + (−3 − 5.19i)13-s + 6·17-s − 4·19-s + (−0.499 + 0.866i)25-s + (−1 + 1.73i)29-s + (4 + 6.92i)31-s − 2·37-s + (−3 − 5.19i)41-s + (−6 + 10.3i)43-s + (4 − 6.92i)47-s + (3.5 + 6.06i)49-s − 6·53-s − 3.99·55-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.603 + 1.04i)11-s + (−0.832 − 1.44i)13-s + 1.45·17-s − 0.917·19-s + (−0.0999 + 0.173i)25-s + (−0.185 + 0.321i)29-s + (0.718 + 1.24i)31-s − 0.328·37-s + (−0.468 − 0.811i)41-s + (−0.914 + 1.58i)43-s + (0.583 − 1.01i)47-s + (0.5 + 0.866i)49-s − 0.824·53-s − 0.539·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(25.8715\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8042966646\)
\(L(\frac12)\) \(\approx\) \(0.8042966646\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6 - 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.910209269918978888702986569809, −8.045604733230375329004763712853, −7.48022522321709425468438581927, −6.85093745763670635652888586003, −5.77736851132505997741793556810, −5.22566176331012922434777711248, −4.40213612816041739260271274480, −3.16836673561107486553005468021, −2.62052270679392534567657702583, −1.39097051699525463643328629306, 0.23843500139855372075214458488, 1.65092101341958065858140906244, 2.60915093873376136705339227485, 3.64814705817298084049408065344, 4.53925202465065590897985155129, 5.32497155985260880778888837747, 6.06288676651208876461685178181, 6.81364482004581152325026080271, 7.76757461116773140042942922589, 8.322873086338119079293919003831

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