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Properties

Label 2-3240-9.4-c1-0-33
Degree $2$
Conductor $3240$
Sign $0.173 + 0.984i$
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 + (−1.68 − 2.92i)7-s + (1.18 + 2.05i)11-s + (1.68 − 2.92i)13-s + 6.74·17-s + 19-s + (2.68 − 4.65i)23-s + (−0.499 − 0.866i)25-s + (1.18 + 2.05i)29-s + (−5.55 + 9.62i)31-s − 3.37·35-s + 6·37-s + (0.127 − 0.221i)41-s + (2.37 + 4.10i)43-s + (−4.68 − 8.11i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.637 − 1.10i)7-s + (0.357 + 0.619i)11-s + (0.467 − 0.809i)13-s + 1.63·17-s + 0.229·19-s + (0.560 − 0.970i)23-s + (−0.0999 − 0.173i)25-s + (0.220 + 0.381i)29-s + (−0.998 + 1.72i)31-s − 0.570·35-s + 0.986·37-s + (0.0199 − 0.0345i)41-s + (0.361 + 0.626i)43-s + (−0.683 − 1.18i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.173 + 0.984i$
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: \((2,\ 3240,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.875237316\)
\(L(\frac12)\) \(\approx\) \(1.875237316\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 2.92i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.18 - 2.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.68 + 2.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (-2.68 + 4.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.18 - 2.05i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.55 - 9.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-0.127 + 0.221i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.37 - 4.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.68 + 8.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.37 + 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.372 + 0.644i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.37T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + (-1.37 - 2.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5 + 8.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 + (-2.37 - 4.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.400077895531641708352019145881, −7.74736746467743313249571357043, −6.96814818203086534525728437085, −6.35099772989000685213767813321, −5.35588359540097145314319695253, −4.69608712032358468060580860442, −3.60148448259644478562198771181, −3.11879955324003492350427039399, −1.55121773493558672883958471402, −0.66615072709403905147191067825, 1.18736198595815857656106018395, 2.39379911977339480225420684982, 3.23919982772485728603187505649, 3.93021461278635452995412584924, 5.23706705582916443126165243521, 5.98208856926853479526461508057, 6.26406453239641901429180780728, 7.41014024188427632536959824248, 8.004662167093226283376283803339, 9.112783802350388738412434397114

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