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Properties

Label 2-2664-37.26-c1-0-43
Degree $2$
Conductor $2664$
Sign $-0.995 - 0.0914i$
Analytic cond. $21.2721$
Root an. cond. $4.61217$
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  + (0.719 − 1.24i)5-s + (−0.5 + 0.866i)7-s + 0.950·11-s + (−0.219 + 0.380i)13-s + (−3.19 − 5.53i)17-s + (−1.91 + 3.31i)19-s − 8.21·23-s + (1.46 + 2.53i)25-s − 6.48·29-s − 3.38·31-s + (0.719 + 1.24i)35-s + (−6.07 + 0.337i)37-s + (3.86 − 6.69i)41-s + 2.43·43-s + 4.48·47-s + ⋯
L(s)  = 1  + (0.321 − 0.557i)5-s + (−0.188 + 0.327i)7-s + 0.286·11-s + (−0.0609 + 0.105i)13-s + (−0.774 − 1.34i)17-s + (−0.439 + 0.760i)19-s − 1.71·23-s + (0.292 + 0.507i)25-s − 1.20·29-s − 0.608·31-s + (0.121 + 0.210i)35-s + (−0.998 + 0.0554i)37-s + (0.603 − 1.04i)41-s + 0.371·43-s + 0.654·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $-0.995 - 0.0914i$
Analytic conductor: \(21.2721\)
Root analytic conductor: \(4.61217\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :1/2),\ -0.995 - 0.0914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1104986835\)
\(L(\frac12)\) \(\approx\) \(0.1104986835\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37 \( 1 + (6.07 - 0.337i)T \)
good5 \( 1 + (-0.719 + 1.24i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.950T + 11T^{2} \)
13 \( 1 + (0.219 - 0.380i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.19 + 5.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.91 - 3.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.21T + 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + 3.38T + 31T^{2} \)
41 \( 1 + (-3.86 + 6.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 2.43T + 43T^{2} \)
47 \( 1 - 4.48T + 47T^{2} \)
53 \( 1 + (1.63 + 2.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.964 - 1.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0497 - 0.0861i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.46 - 4.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.87 - 13.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.29T + 73T^{2} \)
79 \( 1 + (-6.80 + 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.26 + 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.82 + 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.80T + 97T^{2} \)
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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

−8.752439916099930274724478864408, −7.62024986087362540907845021619, −7.03311285151343340906795367895, −5.94018775021437557394343799655, −5.51228972518803165810545768026, −4.46635419629260376547191890871, −3.71988134943253931354863173432, −2.47620685894302330138388286844, −1.61632299791508643472062648684, −0.03287787385786286010583653168, 1.71607485969665544763254247241, 2.55690178525622219475869570425, 3.77264465999852769170473056556, 4.29347470665850535745089104550, 5.53907606059000140905041424047, 6.28740024789598525567160648070, 6.81692945134674261713464439312, 7.71963578263929128748941335011, 8.489717844640176188649458644515, 9.258624087713703480830151483822

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