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Properties

Label 2-3024-252.115-c1-0-45
Degree $2$
Conductor $3024$
Sign $-0.976 + 0.213i$
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  + (0.679 + 0.392i)5-s + (2.49 + 0.891i)7-s + (−1.22 + 0.708i)11-s + (−1.50 + 0.868i)13-s + (−5.43 − 3.13i)17-s + (−0.736 − 1.27i)19-s + (−4.85 − 2.80i)23-s + (−2.19 − 3.79i)25-s + (−3.95 + 6.85i)29-s − 8.41·31-s + (1.34 + 1.58i)35-s + (3.74 + 6.48i)37-s + (−7.19 + 4.15i)41-s + (−7.85 − 4.53i)43-s − 0.110·47-s + ⋯
L(s)  = 1  + (0.303 + 0.175i)5-s + (0.941 + 0.337i)7-s + (−0.369 + 0.213i)11-s + (−0.417 + 0.240i)13-s + (−1.31 − 0.761i)17-s + (−0.168 − 0.292i)19-s + (−1.01 − 0.584i)23-s + (−0.438 − 0.759i)25-s + (−0.734 + 1.27i)29-s − 1.51·31-s + (0.227 + 0.267i)35-s + (0.615 + 1.06i)37-s + (−1.12 + 0.648i)41-s + (−1.19 − 0.691i)43-s − 0.0160·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.976 + 0.213i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.976 + 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02218350205\)
\(L(\frac12)\) \(\approx\) \(0.02218350205\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.49 - 0.891i)T \)
good5 \( 1 + (-0.679 - 0.392i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.22 - 0.708i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.50 - 0.868i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.43 + 3.13i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.736 + 1.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.85 + 2.80i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.95 - 6.85i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.41T + 31T^{2} \)
37 \( 1 + (-3.74 - 6.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.19 - 4.15i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.85 + 4.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.110T + 47T^{2} \)
53 \( 1 + (-4.28 + 7.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.0736T + 59T^{2} \)
61 \( 1 + 1.23iT - 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + 0.390iT - 71T^{2} \)
73 \( 1 + (-3.70 - 2.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.00iT - 79T^{2} \)
83 \( 1 + (-7.88 + 13.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.15 - 3.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.89 - 3.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.451158423946043100390794961418, −7.62738074238560599746901430042, −6.87518365990420930304791184506, −6.15697013680510899814159098274, −5.02458724488552407146005001350, −4.77399596376175688950537843250, −3.58675223181114768945927139448, −2.32800620631114856109446663355, −1.86312708157227912992401932141, −0.00608471482972038239556773382, 1.66514827245858969672421301231, 2.26342815696956513047538798839, 3.74736242026487412792024370692, 4.30614408082986663841031737357, 5.40661626551802141930960880861, 5.80051120878378850319465701224, 6.92383307206548306516316868233, 7.68845393993869919081369839639, 8.228531166371912336410767592637, 9.033039529451401934809047900519

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