Properties

Degree $2$
Conductor $3024$
Sign $-0.999 + 0.0286i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.234 − 0.405i)5-s + (−0.212 + 2.63i)7-s + (0.674 − 1.16i)11-s + (−3.16 + 5.48i)13-s + (2.47 + 4.28i)17-s + (−2.38 + 4.13i)19-s + (−3.81 − 6.60i)23-s + (2.39 − 4.14i)25-s + (1.80 + 3.12i)29-s − 6.49·31-s + (1.11 − 0.531i)35-s + (5.24 − 9.07i)37-s + (0.0251 − 0.0435i)41-s + (0.431 + 0.748i)43-s − 10.9·47-s + ⋯
L(s)  = 1  + (−0.104 − 0.181i)5-s + (−0.0802 + 0.996i)7-s + (0.203 − 0.352i)11-s + (−0.877 + 1.52i)13-s + (0.599 + 1.03i)17-s + (−0.548 + 0.949i)19-s + (−0.795 − 1.37i)23-s + (0.478 − 0.828i)25-s + (0.335 + 0.580i)29-s − 1.16·31-s + (0.189 − 0.0897i)35-s + (0.861 − 1.49i)37-s + (0.00392 − 0.00680i)41-s + (0.0658 + 0.114i)43-s − 1.60·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.999 + 0.0286i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.999 + 0.0286i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4148849414\)
\(L(\frac12)\) \(\approx\) \(0.4148849414\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.212 - 2.63i)T \)
good5 \( 1 + (0.234 + 0.405i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.674 + 1.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.16 - 5.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.47 - 4.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.38 - 4.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.81 + 6.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.80 - 3.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
37 \( 1 + (-5.24 + 9.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0251 + 0.0435i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.431 - 0.748i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + (5.84 + 10.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.87T + 59T^{2} \)
61 \( 1 - 3.74T + 61T^{2} \)
67 \( 1 - 2.64T + 67T^{2} \)
71 \( 1 + 7.04T + 71T^{2} \)
73 \( 1 + (3.30 + 5.71i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 3.17T + 79T^{2} \)
83 \( 1 + (-4.90 - 8.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.30 - 9.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.97 - 12.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.010919565398452429293434328416, −8.417680839257516128059867058265, −7.78407284271996421163443478682, −6.57105678722778860105679048241, −6.24255945131412700991126475258, −5.27840284585627143403972050023, −4.40560131949218946925899351219, −3.65920381456461291938112006607, −2.40771111003729067531120379322, −1.72327616797851140902084011823, 0.12706093161228556230836122383, 1.35691704097806231735071908758, 2.80675995991440222206585973863, 3.40124891302490115270190337986, 4.52886596590345144449929582323, 5.14321565028762578240812029209, 6.07439339735577766606131297664, 7.11233365263341587114873010643, 7.49930711838290071017040736539, 8.095965806183168078156641890490

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