Properties

Degree $2$
Conductor $3240$
Sign $-0.766 - 0.642i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $1$

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: $-0.766 - 0.642i$
Motivic weight: \(1\)
Character: $\chi_{3240} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−8.242155141583757194759864805876, −7.62911776757967538617663112329, −6.50321884540174078969104703187, −6.10449497952573634087183106523, −4.95848890397936305559578185465, −4.45905685587002289731098115620, −3.32764672861965829004093677583, −2.58676221891270197606603675293, −1.19723752386440659352610619767, 0, 1.95014402850216305125437303690, 2.35199065601340311288318228898, 3.91663436435070860240344292739, 4.31585622469703807855010308937, 5.14889466791944335456037666305, 6.48506926316086115192751231418, 6.78629005098767373322008202528, 7.40600431903669641257903946210, 8.499466376583209059613860942847

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