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Properties

Label 2-3024-63.25-c1-0-45
Degree $2$
Conductor $3024$
Sign $-0.928 + 0.370i$
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  + (0.880 − 1.52i)5-s + (0.710 − 2.54i)7-s + (−3.06 − 5.30i)11-s + (−0.380 − 0.658i)13-s + (3.42 − 5.92i)17-s + (−0.971 − 1.68i)19-s + (0.210 − 0.364i)23-s + (0.949 + 1.64i)25-s + (−0.732 + 1.26i)29-s − 7.70·31-s + (−3.26 − 3.32i)35-s + (1.44 + 2.49i)37-s + (3.47 + 6.01i)41-s + (−4.33 + 7.49i)43-s + 1.66·47-s + ⋯
L(s)  = 1  + (0.393 − 0.681i)5-s + (0.268 − 0.963i)7-s + (−0.923 − 1.59i)11-s + (−0.105 − 0.182i)13-s + (0.829 − 1.43i)17-s + (−0.222 − 0.385i)19-s + (0.0438 − 0.0760i)23-s + (0.189 + 0.328i)25-s + (−0.135 + 0.235i)29-s − 1.38·31-s + (−0.551 − 0.562i)35-s + (0.237 + 0.410i)37-s + (0.542 + 0.939i)41-s + (−0.660 + 1.14i)43-s + 0.242·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.928 + 0.370i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.928 + 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.423189284\)
\(L(\frac12)\) \(\approx\) \(1.423189284\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.710 + 2.54i)T \)
good5 \( 1 + (-0.880 + 1.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.06 + 5.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.380 + 0.658i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.42 + 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.971 + 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.210 + 0.364i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.732 - 1.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.70T + 31T^{2} \)
37 \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.47 - 6.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.33 - 7.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.66T + 47T^{2} \)
53 \( 1 + (-0.112 + 0.195i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.98T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 6.78T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (-0.153 + 0.265i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + (1.56 - 2.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.81 - 3.14i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.274995097964969264853297746553, −7.76582037259915606928983287523, −6.99782532623571970352784503606, −5.95774442825453980185795004741, −5.23497208488002171356279249519, −4.71700993505913510973139705953, −3.47640640673052778665665831283, −2.80814779099609589176940875045, −1.31435754173717665599972443421, −0.44250443879081540194588754877, 1.89409424381947865001252320870, 2.24167582731249216918837775677, 3.41988914194163612634710764805, 4.42585138972749456960606413994, 5.39741830177932404390273580140, 5.88649273344375571267408437675, 6.83410091972145345309397185266, 7.57600777234739210851460261038, 8.208443847418284319266645465600, 9.123812191036967069107107041168

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