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Properties

Label 2-3024-63.25-c1-0-29
Degree $2$
Conductor $3024$
Sign $0.483 + 0.875i$
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  + (−1.26 + 2.18i)5-s + (−0.527 + 2.59i)7-s + (0.687 + 1.18i)11-s + (−2.80 − 4.84i)13-s + (2.69 − 4.66i)17-s + (−2.44 − 4.23i)19-s + (−2.08 + 3.61i)23-s + (−0.675 − 1.17i)25-s + (1.56 − 2.71i)29-s − 4.80·31-s + (−4.99 − 4.41i)35-s + (−2.69 − 4.67i)37-s + (3.02 + 5.24i)41-s + (−2.44 + 4.23i)43-s − 5.65·47-s + ⋯
L(s)  = 1  + (−0.563 + 0.976i)5-s + (−0.199 + 0.979i)7-s + (0.207 + 0.358i)11-s + (−0.776 − 1.34i)13-s + (0.653 − 1.13i)17-s + (−0.561 − 0.972i)19-s + (−0.435 + 0.753i)23-s + (−0.135 − 0.234i)25-s + (0.291 − 0.504i)29-s − 0.862·31-s + (−0.844 − 0.746i)35-s + (−0.443 − 0.768i)37-s + (0.473 + 0.819i)41-s + (−0.373 + 0.646i)43-s − 0.824·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.483 + 0.875i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8665770364\)
\(L(\frac12)\) \(\approx\) \(0.8665770364\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.527 - 2.59i)T \)
good5 \( 1 + (1.26 - 2.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.687 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.80 + 4.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.69 + 4.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.44 + 4.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.08 - 3.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.56 + 2.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.80T + 31T^{2} \)
37 \( 1 + (2.69 + 4.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.02 - 5.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.44 - 4.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + (-7.00 + 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 + 6.85T + 61T^{2} \)
67 \( 1 - 8.11T + 67T^{2} \)
71 \( 1 + 2.25T + 71T^{2} \)
73 \( 1 + (-3.51 + 6.08i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.75T + 79T^{2} \)
83 \( 1 + (-7.48 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.75 + 4.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.894 + 1.54i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.525988772635669814514101475587, −7.69574584758276595998730780649, −7.20855966857783125093747117658, −6.41189244184309399659268538456, −5.45840834845268175703665339553, −4.89155958976987495783481394215, −3.57549557203664177934998931286, −2.93644222318197337133683704132, −2.19143066350705618119380310332, −0.31106503737810961276982083844, 1.02728141802279663536581089559, 2.03947487492716966111298214497, 3.63618381217619511934441062535, 4.08415867264161564124846404609, 4.79269188227308692101260103195, 5.81241668684645859085210680120, 6.69695094531698985085856080253, 7.34522021312087575323885722215, 8.294266312781391320028941661644, 8.614241662260382940199981197425

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