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Properties

Label 2-3240-9.4-c1-0-24
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 + (1 + 1.73i)11-s − 3·17-s − 19-s + (1.5 − 2.59i)23-s + (−0.499 − 0.866i)25-s + (2 + 3.46i)29-s + (2.5 − 4.33i)31-s + 10·37-s + (3 − 5.19i)41-s + (3 + 5.19i)43-s + (4 + 6.92i)47-s + (3.5 − 6.06i)49-s − 3·53-s + 1.99·55-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.301 + 0.522i)11-s − 0.727·17-s − 0.229·19-s + (0.312 − 0.541i)23-s + (−0.0999 − 0.173i)25-s + (0.371 + 0.643i)29-s + (0.449 − 0.777i)31-s + 1.64·37-s + (0.468 − 0.811i)41-s + (0.457 + 0.792i)43-s + (0.583 + 1.01i)47-s + (0.5 − 0.866i)49-s − 0.412·53-s + 0.269·55-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

\(L(1)\) \(\approx\) \(1.918067036\)
\(L(\frac12)\) \(\approx\) \(1.918067036\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
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

−8.633640529368152430870388309197, −7.941890736681617848041627856411, −7.06842397019783576798061920777, −6.37144729626800161346533947602, −5.62104538425716082708062795148, −4.58796546941800055053733911465, −4.18099065835868493197725825981, −2.86391029664859266550288372116, −2.01146102515500321100595290234, −0.78389592068946060390318130506, 0.904509761382959684226773690652, 2.20541625893556350196532544586, 3.03631727904858194101487133208, 4.01439755456307918209020040734, 4.80259045979508856298416584249, 5.86094092563580165313679689839, 6.35634207652174261122049187729, 7.18436131375254493863215124108, 7.935129861964059786598461546339, 8.771845772725393772005347264934

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