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Properties

Label 2-3240-360.149-c0-0-7
Degree $2$
Conductor $3240$
Sign $0.342 - 0.939i$
Analytic cond. $1.61697$
Root an. cond. $1.27160$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + (0.866 + 0.5i)5-s − i·8-s + (−0.5 + 0.866i)10-s + 16-s + 1.73·17-s − 1.73i·19-s + (−0.866 − 0.5i)20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + i·32-s + 1.73i·34-s + 1.73·38-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (0.866 + 0.5i)5-s − i·8-s + (−0.5 + 0.866i)10-s + 16-s + 1.73·17-s − 1.73i·19-s + (−0.866 − 0.5i)20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + i·32-s + 1.73i·34-s + 1.73·38-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404545203\)
\(L(\frac12)\) \(\approx\) \(1.404545203\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-0.866 - 0.5i)T \)
good7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.73T + T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.060088547771569945979543680887, −8.148618518599986364943368090415, −7.30411786431986310336092726877, −6.74400315288030076761473052846, −6.09473027846842103943806862673, −5.22042247191546266071876571888, −4.74455852737514805773542366627, −3.44127360169184307980857327107, −2.66708967435778655997564908732, −1.10769528209305279792276205124, 1.22515391252203919489691505681, 1.86352375353004977644902132480, 3.12749700554314818024171032123, 3.74998324164523930779071634949, 4.86607119607262878619061869337, 5.58945419719353814302194953542, 6.01111114201036442244783694400, 7.50758576045833287609677709094, 8.050309572068750927354834773217, 8.939206964550131587024051135227

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