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Properties

Label 2-3240-9.4-c1-0-10
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 + (−1 − 1.73i)7-s + (−1 − 1.73i)11-s + (−2 + 3.46i)13-s − 2·17-s + 4·19-s + (−4 + 6.92i)23-s + (−0.499 − 0.866i)25-s + (5 + 8.66i)29-s + (−2 + 3.46i)31-s − 1.99·35-s + (4 + 6.92i)43-s + (−4 − 6.92i)47-s + (1.50 − 2.59i)49-s + 6·53-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.377 − 0.654i)7-s + (−0.301 − 0.522i)11-s + (−0.554 + 0.960i)13-s − 0.485·17-s + 0.917·19-s + (−0.834 + 1.44i)23-s + (−0.0999 − 0.173i)25-s + (0.928 + 1.60i)29-s + (−0.359 + 0.622i)31-s − 0.338·35-s + (0.609 + 1.05i)43-s + (−0.583 − 1.01i)47-s + (0.214 − 0.371i)49-s + 0.824·53-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: $\chi_{3240} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.434164813\)
\(L(\frac12)\) \(\approx\) \(1.434164813\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
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 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5 - 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
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 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.751966682090307361619963298816, −8.035831134859649727814558178941, −7.07466177252173673408701584780, −6.71500434826729664497219941105, −5.54999970868903942184117146830, −5.02271689371519504678249049483, −3.98465489816492985488276519509, −3.27728285415042163302564604477, −2.08531394920969656021366679698, −0.986983704824496728386085548899, 0.51449936036839256445729010830, 2.34529910652283620568222327583, 2.61011180987564294966991481912, 3.84801435718473393203134972131, 4.78020540398903541062692228778, 5.63856445524782068522381725197, 6.23843218956942996894456823101, 7.07985902831138621634707407752, 7.86091629280777629414249191508, 8.479790118189527332872079802281

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