Properties

Degree $2$
Conductor $3024$
Sign $-0.997 + 0.0717i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.667 + 1.15i)5-s + (−1.90 + 1.83i)7-s + (−0.756 + 1.31i)11-s + (−2.58 + 4.48i)13-s + (−0.774 − 1.34i)17-s + (1.25 − 2.16i)19-s + (3.68 + 6.37i)23-s + (1.60 − 2.78i)25-s + (0.0309 + 0.0536i)29-s + 3.84·31-s + (−3.39 − 0.972i)35-s + (−0.281 + 0.487i)37-s + (−4.51 + 7.81i)41-s + (−5.09 − 8.83i)43-s − 9.51·47-s + ⋯
L(s)  = 1  + (0.298 + 0.516i)5-s + (−0.719 + 0.694i)7-s + (−0.228 + 0.395i)11-s + (−0.717 + 1.24i)13-s + (−0.187 − 0.325i)17-s + (0.287 − 0.497i)19-s + (0.767 + 1.32i)23-s + (0.321 − 0.557i)25-s + (0.00575 + 0.00996i)29-s + 0.691·31-s + (−0.573 − 0.164i)35-s + (−0.0462 + 0.0801i)37-s + (−0.704 + 1.22i)41-s + (−0.777 − 1.34i)43-s − 1.38·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.997 + 0.0717i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.997 + 0.0717i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6409333778\)
\(L(\frac12)\) \(\approx\) \(0.6409333778\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.90 - 1.83i)T \)
good5 \( 1 + (-0.667 - 1.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.756 - 1.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.58 - 4.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.774 + 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.25 + 2.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.68 - 6.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0309 - 0.0536i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 + (0.281 - 0.487i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.51 - 7.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.51T + 47T^{2} \)
53 \( 1 + (0.755 + 1.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.44T + 59T^{2} \)
61 \( 1 - 3.23T + 61T^{2} \)
67 \( 1 + 6.93T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (1.37 + 2.38i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.91T + 79T^{2} \)
83 \( 1 + (-2.80 - 4.85i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.703 - 1.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.09 + 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.257399977681545363211150948968, −8.487127572438050877645234548200, −7.39059669962354215241947877350, −6.80986738568436506797183577199, −6.26706760723486418826328154811, −5.20248294361210544827052155288, −4.60630090611872554440608817876, −3.32062032638712667525191290594, −2.66013878914585136660806809303, −1.71638982685601846014052586001, 0.19907604573212221869333258266, 1.30359057440518970210895709350, 2.77597077607806129164028570514, 3.38437378079658750865493840989, 4.55385225951294334690230237093, 5.20355810743133343630330083480, 6.08019461084635111363431967398, 6.79737368144474693254514315062, 7.65390120265058385172514934610, 8.317060162421387311085046642178

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