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Properties

Label 2-3240-9.7-c1-0-42
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 + (1.63 − 2.83i)7-s + (3.13 − 5.43i)11-s + (0.637 + 1.10i)13-s − 2·17-s + 19-s + (−3.63 − 6.30i)23-s + (−0.499 + 0.866i)25-s + (3.13 − 5.43i)29-s + (−3.13 − 5.43i)31-s + 3.27·35-s − 10.5·37-s + (3.77 + 6.53i)41-s + (−2 + 3.46i)43-s + (−0.637 + 1.10i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.618 − 1.07i)7-s + (0.945 − 1.63i)11-s + (0.176 + 0.306i)13-s − 0.485·17-s + 0.229·19-s + (−0.758 − 1.31i)23-s + (−0.0999 + 0.173i)25-s + (0.582 − 1.00i)29-s + (−0.563 − 0.976i)31-s + 0.553·35-s − 1.73·37-s + (0.589 + 1.02i)41-s + (−0.304 + 0.528i)43-s + (−0.0929 + 0.161i)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: $\chi_{3240} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.828331913\)
\(L(\frac12)\) \(\approx\) \(1.828331913\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 2.83i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.13 + 5.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.637 - 1.10i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (3.63 + 6.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.13 + 5.43i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.13 + 5.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + (-3.77 - 6.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.637 - 1.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.725T + 53T^{2} \)
59 \( 1 + (6.5 + 11.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.27 - 7.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.274 + 0.476i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.27T + 71T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 + (5.27 - 9.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.27 - 2.20i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + (8 - 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.282599880241827661087792557213, −7.898706578980911183107907815168, −6.67708587768542008146231357895, −6.44439161638244756221809890450, −5.48526199359898505932658830444, −4.33132449141923897721992982461, −3.87182890914691739407691516110, −2.84031971562717863385152752466, −1.62731528260964336680439519724, −0.55141641644469380096231791635, 1.58696102562677015180927534742, 1.98754295569420249543457767860, 3.33898244390838081670635081296, 4.32113305648733052038242482468, 5.15035540016763609335188534707, 5.61433079347366515555313921609, 6.73252527439025028725544540354, 7.28493773686978297819117615875, 8.254979405553045826877212491352, 8.969835319612413228714829327110

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