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Properties

Label 2-3024-252.223-c1-0-34
Degree $2$
Conductor $3024$
Sign $0.513 + 0.858i$
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 + (−0.5 − 2.59i)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 + (−6.5 + 2.59i)49-s + 6·53-s + 6·55-s + ⋯
L(s)  = 1  + (0.670 − 0.387i)5-s + (−0.188 − 0.981i)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 + (−0.928 + 0.371i)49-s + 0.824·53-s + 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.513 + 0.858i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.513 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.243646802\)
\(L(\frac12)\) \(\approx\) \(2.243646802\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.59i)T \)
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

−8.945683414829997404139062620914, −7.62727300640102502468228443353, −7.11708778832760975771970914981, −6.46888852627793628715580210773, −5.41586452223043522589856572743, −4.79280298185888414993607324861, −3.87213903421372306128615619943, −2.98869598822742395921994901111, −1.69632711241909373685847812205, −0.804519022058974511771790973598, 1.21627390020321927106706124083, 2.28261020180788154400970575992, 3.16715704163805757489604904340, 3.99631125450544176615577517898, 5.31794396289941992209222986347, 5.74455806487876971739317614612, 6.52383848153931433298619184336, 7.18141404813543867094103954358, 8.308027759290076081986289554446, 8.850403822183711076614661473496

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