Properties

Degree $2$
Conductor $3024$
Sign $0.889 - 0.457i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.66·5-s + (−1.94 + 1.79i)7-s + 1.36·11-s + (−2.75 − 4.77i)13-s + (1.23 + 2.14i)17-s + (2.19 − 3.80i)19-s − 4.69·23-s + 2.11·25-s + (−2.94 + 5.10i)29-s + (1.55 − 2.69i)31-s + (5.18 − 4.78i)35-s + (−3.15 + 5.46i)37-s + (−1.38 − 2.40i)41-s + (4.87 − 8.45i)43-s + (5.02 + 8.70i)47-s + ⋯
L(s)  = 1  − 1.19·5-s + (−0.735 + 0.678i)7-s + 0.411·11-s + (−0.764 − 1.32i)13-s + (0.300 + 0.520i)17-s + (0.503 − 0.872i)19-s − 0.977·23-s + 0.423·25-s + (−0.547 + 0.948i)29-s + (0.279 − 0.484i)31-s + (0.877 − 0.809i)35-s + (−0.518 + 0.898i)37-s + (−0.216 − 0.375i)41-s + (0.744 − 1.28i)43-s + (0.732 + 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.889 - 0.457i$
Motivic weight: \(1\)
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.889 - 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9220281858\)
\(L(\frac12)\) \(\approx\) \(0.9220281858\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.94 - 1.79i)T \)
good5 \( 1 + 2.66T + 5T^{2} \)
11 \( 1 - 1.36T + 11T^{2} \)
13 \( 1 + (2.75 + 4.77i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.23 - 2.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.19 + 3.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.69T + 23T^{2} \)
29 \( 1 + (2.94 - 5.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.55 + 2.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.15 - 5.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.38 + 2.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.87 + 8.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.02 - 8.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.47 - 2.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.77 - 3.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.663 + 1.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.14 + 7.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (1.11 + 1.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.41 - 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.15 + 8.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.73 - 13.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.55 - 4.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.749408193970082168341381408734, −7.919875382809976950271128476078, −7.45196393497727591171444095039, −6.56489412932711502538275606325, −5.68491731935751089317359790450, −4.97274928870538106326137232534, −3.86172810142128299303556613809, −3.28287734823425944673092073459, −2.34675313772667827322958967622, −0.64909318151157525249944437196, 0.49375826778517289391403270543, 1.96834917349686155074034342265, 3.27006735745125026084041588264, 3.98991754036929950541466554304, 4.44987986682232378993834435912, 5.66442864717076557284234078843, 6.59447515784731922740940979693, 7.26846512713287069132677721465, 7.73297185863930349198022081993, 8.593680584560972016872310284534

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