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Properties

Label 2-3024-252.31-c1-0-15
Degree $2$
Conductor $3024$
Sign $-0.721 - 0.691i$
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  + 2.85i·5-s + (−1.73 + 2.00i)7-s + 2.26i·11-s + (5.59 + 3.22i)13-s + (4.74 + 2.74i)17-s + (1.59 + 2.75i)19-s + 5.97i·23-s − 3.16·25-s + (−1.06 − 1.85i)29-s + (−2.01 − 3.49i)31-s + (−5.71 − 4.94i)35-s + (−5.06 − 8.77i)37-s + (9.02 + 5.20i)41-s + (0.0397 − 0.0229i)43-s + (2.87 − 4.97i)47-s + ⋯
L(s)  = 1  + 1.27i·5-s + (−0.654 + 0.756i)7-s + 0.681i·11-s + (1.55 + 0.895i)13-s + (1.15 + 0.664i)17-s + (0.365 + 0.632i)19-s + 1.24i·23-s − 0.632·25-s + (−0.198 − 0.343i)29-s + (−0.362 − 0.627i)31-s + (−0.966 − 0.835i)35-s + (−0.832 − 1.44i)37-s + (1.40 + 0.813i)41-s + (0.00605 − 0.00349i)43-s + (0.419 − 0.725i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.721 - 0.691i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1711, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.721 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.819468786\)
\(L(\frac12)\) \(\approx\) \(1.819468786\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 2.00i)T \)
good5 \( 1 - 2.85iT - 5T^{2} \)
11 \( 1 - 2.26iT - 11T^{2} \)
13 \( 1 + (-5.59 - 3.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.74 - 2.74i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.59 - 2.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.97iT - 23T^{2} \)
29 \( 1 + (1.06 + 1.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.01 + 3.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.06 + 8.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.02 - 5.20i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0397 + 0.0229i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.87 + 4.97i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.43 + 4.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.10 - 5.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.05 + 3.49i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.82 + 3.94i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.5iT - 71T^{2} \)
73 \( 1 + (-11.0 - 6.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.4 + 6.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.88 + 3.26i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.95 + 2.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.62 - 2.67i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.154020993818553576598881865496, −8.196612977666159231160255521116, −7.39663007526690352227649946580, −6.74234283836200619941516233491, −5.92428812741166432055598227089, −5.56858374369765801622147058095, −3.86011576822138006181555200722, −3.60824766349972890042154180805, −2.50167206403225964807587739645, −1.54675499209118107289505072711, 0.68754388134770576042656776234, 1.14443711856057873391258043191, 2.97718773805792632689994537803, 3.59902734756140634725514161513, 4.57224657452436714094725435240, 5.39181860344308157564605269798, 6.04159469965104153519854211595, 6.93552915452917363520793694986, 7.84944684279162674714455487707, 8.507427467945527357172330680929

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