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Properties

Label 2-3024-252.139-c1-0-34
Degree $2$
Conductor $3024$
Sign $-0.158 + 0.987i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
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  + (−1.5 − 0.866i)5-s + (2.5 + 0.866i)7-s + (−3 + 1.73i)11-s − 3.46i·17-s + 5·19-s + (−4.5 − 2.59i)23-s + (−1 − 1.73i)25-s + (1 − 1.73i)31-s + (−3 − 3.46i)35-s − 8·37-s + (−3 + 1.73i)43-s + (3 + 5.19i)47-s + (5.5 + 4.33i)49-s + 6·53-s + 6·55-s + ⋯
L(s)  = 1  + (−0.670 − 0.387i)5-s + (0.944 + 0.327i)7-s + (−0.904 + 0.522i)11-s − 0.840i·17-s + 1.14·19-s + (−0.938 − 0.541i)23-s + (−0.200 − 0.346i)25-s + (0.179 − 0.311i)31-s + (−0.507 − 0.585i)35-s − 1.31·37-s + (−0.457 + 0.264i)43-s + (0.437 + 0.757i)47-s + (0.785 + 0.618i)49-s + 0.824·53-s + 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.158 + 0.987i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.158 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.121823401\)
\(L(\frac12)\) \(\approx\) \(1.121823401\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (4.5 + 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 - 1.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 3.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + (-13.5 + 7.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 + (9 - 5.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.356133329880255406253800482812, −7.78958486014594410131458167500, −7.32403717650584159299234719550, −6.18366078541342110186705556429, −5.13438099244516297522625383856, −4.84068769042852442366852735006, −3.85685039482281226472203193899, −2.74399898355555187323051042182, −1.79610924846640378728027785163, −0.37601123836633134587383584250, 1.19293367143914383277498960945, 2.37256001879717717481546646431, 3.51786932857845135826133375004, 4.04936060396098000100790647251, 5.26932767790035450503310112704, 5.62022948200431763442137779160, 6.93391765052731645493305471963, 7.47447159397525095552671905370, 8.179525854712254095318755138019, 8.610078189063965862907269482288

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