Properties

Label 2-3240-5.4-c1-0-23
Degree $2$
Conductor $3240$
Sign $0.894 - 0.447i$
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  + (−1 − 2i)5-s − 2i·7-s + 5·11-s + 4i·13-s + 6i·17-s + 7·19-s + 4i·23-s + (−3 + 4i)25-s − 5·29-s − 3·31-s + (−4 + 2i)35-s + 2i·37-s − 7·41-s + 6i·43-s + 6i·47-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s − 0.755i·7-s + 1.50·11-s + 1.10i·13-s + 1.45i·17-s + 1.60·19-s + 0.834i·23-s + (−0.600 + 0.800i)25-s − 0.928·29-s − 0.538·31-s + (−0.676 + 0.338i)35-s + 0.328i·37-s − 1.09·41-s + 0.914i·43-s + 0.875i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(25.8715\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.731452527\)
\(L(\frac12)\) \(\approx\) \(1.731452527\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 15T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.751247200358200256677502074403, −7.973103881985362746939187645620, −7.23408295871254035710001612488, −6.56678118655913522222147313566, −5.64194709881183865967712858245, −4.75515824880593334908041869324, −3.74976450356557121483933309561, −3.72212784264987148652800031930, −1.68918136824575440729859233700, −1.15113616788828396308145435499, 0.61838830697776382225402053268, 2.10778927255725965753624548073, 3.14899033437529600440979433568, 3.59387681436439858374163140986, 4.82508550326625055024023385626, 5.61749562875058607116179341784, 6.37484696475546100397345377836, 7.25816312128100823202082167354, 7.59925169840168080336403985486, 8.739766977327316168551549030044

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