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Properties

Label 2-3240-9.4-c1-0-4
Degree $2$
Conductor $3240$
Sign $-0.939 - 0.342i$
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)7-s + (−1 − 1.73i)11-s + (−2 + 3.46i)13-s + 17-s − 5·19-s + (−2.5 + 4.33i)23-s + (−0.499 − 0.866i)25-s + (−4 − 6.92i)29-s + (−3.5 + 6.06i)31-s + 3.99·35-s − 6·37-s + (−3 + 5.19i)41-s + (1 + 1.73i)43-s + (−4 − 6.92i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.755 + 1.30i)7-s + (−0.301 − 0.522i)11-s + (−0.554 + 0.960i)13-s + 0.242·17-s − 1.14·19-s + (−0.521 + 0.902i)23-s + (−0.0999 − 0.173i)25-s + (−0.742 − 1.28i)29-s + (−0.628 + 1.08i)31-s + 0.676·35-s − 0.986·37-s + (−0.468 + 0.811i)41-s + (0.152 + 0.264i)43-s + (−0.583 − 1.01i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(25.8715\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6471400489\)
\(L(\frac12)\) \(\approx\) \(0.6471400489\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (2.5 - 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (4.5 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.5 - 14.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-4 - 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.904140330653818337165178648858, −8.367117944663208171186397838457, −7.66903149009137062757434750620, −6.63579768194865165219740901249, −5.83065703710561033975219988472, −5.23676594262831427400264063527, −4.51834540101091665565037881762, −3.44867414393349377751825320280, −2.21756807244551354036819310769, −1.73362831597264875195716787381, 0.17931063121907144766927808265, 1.59253611192481790698860195029, 2.53461070912300990198204588456, 3.68260359235524166952283971697, 4.41078073699718774615675132324, 5.20294727933758717150986698979, 6.04419303541784649137779314547, 7.17073448359366060728582154545, 7.36661213231094987205221661183, 8.216014257130467112720472354466

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