Properties

Degree $2$
Conductor $3024$
Sign $0.890 + 0.455i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.89 + 3.29i)5-s + (0.841 + 2.50i)7-s + (−2.25 − 3.90i)11-s + (0.588 + 1.01i)13-s + (2.95 − 5.12i)17-s + (−2.55 − 4.42i)19-s + (2.09 − 3.62i)23-s + (−4.71 − 8.17i)25-s + (−2.11 + 3.65i)29-s + 6.24·31-s + (−9.85 − 1.99i)35-s + (−3.87 − 6.70i)37-s + (−0.754 − 1.30i)41-s + (5.01 − 8.68i)43-s − 2.23·47-s + ⋯
L(s)  = 1  + (−0.849 + 1.47i)5-s + (0.318 + 0.948i)7-s + (−0.680 − 1.17i)11-s + (0.163 + 0.282i)13-s + (0.717 − 1.24i)17-s + (−0.586 − 1.01i)19-s + (0.435 − 0.755i)23-s + (−0.943 − 1.63i)25-s + (−0.392 + 0.679i)29-s + 1.12·31-s + (−1.66 − 0.337i)35-s + (−0.636 − 1.10i)37-s + (−0.117 − 0.204i)41-s + (0.765 − 1.32i)43-s − 0.326·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.890 + 0.455i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.169862602\)
\(L(\frac12)\) \(\approx\) \(1.169862602\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.841 - 2.50i)T \)
good5 \( 1 + (1.89 - 3.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.25 + 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.588 - 1.01i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.55 + 4.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.09 + 3.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.11 - 3.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + (3.87 + 6.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.754 + 1.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.01 + 8.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 + (6.49 - 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 1.45T + 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + (-3.72 + 6.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.84T + 79T^{2} \)
83 \( 1 + (0.307 - 0.532i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.25 - 2.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.36 + 4.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.625548501005726406690833588613, −7.87682044667142738557073437029, −7.15736048742315457817016441116, −6.52341524922827218642071463946, −5.64113112142430933918159772005, −4.86584051230399438640361224042, −3.70764036006880862063224658461, −2.87268733763798103488448030679, −2.45012728281936564420824733352, −0.44664320054469224391619433393, 0.992866727924142115121689944754, 1.81116505435379924143944624921, 3.48808723832897656986943441968, 4.16845630859806376778817041413, 4.79128241699858954568549785639, 5.49122397870650086248426613147, 6.59387469180685487876292071802, 7.60979021904604192754618812550, 8.136486617530177030445853768969, 8.348881116614174778552773196184

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