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Properties

Label 2-3240-45.7-c0-0-1
Degree $2$
Conductor $3240$
Sign $-0.395 + 0.918i$
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  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s − i·19-s + (−1.36 − 0.366i)23-s + (−0.499 + 0.866i)25-s + (−0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + (1 + i)37-s + (−0.5 − 0.866i)41-s + (0.366 + 1.36i)43-s + (1.36 − 0.366i)47-s + (−0.866 + 0.5i)49-s − 0.999·55-s + (0.866 − 0.5i)59-s + (0.366 − 1.36i)67-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s − i·19-s + (−1.36 − 0.366i)23-s + (−0.499 + 0.866i)25-s + (−0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + (1 + i)37-s + (−0.5 − 0.866i)41-s + (0.366 + 1.36i)43-s + (1.36 − 0.366i)47-s + (−0.866 + 0.5i)49-s − 0.999·55-s + (0.866 − 0.5i)59-s + (0.366 − 1.36i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-0.395 + 0.918i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ -0.395 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8958117453\)
\(L(\frac12)\) \(\approx\) \(0.8958117453\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 - iT - T^{2} \)
97 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−8.562974126263168677377767141983, −7.971861923027654664776312939314, −7.24846867333371902245865599202, −6.18051194999892312251386243138, −5.66308496083679596526882974774, −4.57535503067797371118223749568, −4.05377825647364990900376428741, −3.07486326986874691140180639722, −1.84061741166326289224203085376, −0.53625264444602231316939396913, 1.64617204808699591087564462390, 2.59145054899815787886247648445, 3.82059737180627468064527937255, 4.05959512593473052551134817419, 5.38543587721955704290482576483, 6.09806237835353775566729246242, 7.00705637822036041831670112471, 7.45299594030947928545642884728, 8.205358957083760194709144653020, 9.081123070563523181611973273389

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