Properties

Degree $2$
Conductor $3024$
Sign $-0.977 - 0.211i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.263 + 0.455i)5-s + (0.333 − 2.62i)7-s + (−2.30 + 3.99i)11-s + (0.244 − 0.423i)13-s + (−2.75 − 4.77i)17-s + (−1.83 + 3.18i)19-s + (0.0269 + 0.0467i)23-s + (2.36 − 4.09i)25-s + (3.28 + 5.68i)29-s − 6.07·31-s + (1.28 − 0.538i)35-s + (0.223 − 0.387i)37-s + (−2.52 + 4.36i)41-s + (−2.84 − 4.93i)43-s − 9.19·47-s + ⋯
L(s)  = 1  + (0.117 + 0.203i)5-s + (0.125 − 0.992i)7-s + (−0.695 + 1.20i)11-s + (0.0678 − 0.117i)13-s + (−0.668 − 1.15i)17-s + (−0.421 + 0.730i)19-s + (0.00562 + 0.00974i)23-s + (0.472 − 0.818i)25-s + (0.609 + 1.05i)29-s − 1.09·31-s + (0.216 − 0.0910i)35-s + (0.0367 − 0.0637i)37-s + (−0.394 + 0.682i)41-s + (−0.434 − 0.752i)43-s − 1.34·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.977 - 0.211i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09369616003\)
\(L(\frac12)\) \(\approx\) \(0.09369616003\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.333 + 2.62i)T \)
good5 \( 1 + (-0.263 - 0.455i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.30 - 3.99i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.244 + 0.423i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.75 + 4.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.83 - 3.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0269 - 0.0467i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.28 - 5.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.07T + 31T^{2} \)
37 \( 1 + (-0.223 + 0.387i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.52 - 4.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.84 + 4.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.19T + 47T^{2} \)
53 \( 1 + (-4.37 - 7.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.63T + 59T^{2} \)
61 \( 1 + 0.465T + 61T^{2} \)
67 \( 1 + 5.19T + 67T^{2} \)
71 \( 1 - 1.76T + 71T^{2} \)
73 \( 1 + (5.23 + 9.07i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + (-4.49 - 7.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.05 + 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.22 - 9.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.096775320873209117478100916292, −8.244645628465707908323680914615, −7.38998278398079014012346891151, −7.02195027828143267037847938351, −6.17092818600782401391666881252, −4.94360553940222921179209263648, −4.60990713382670062936653048210, −3.51893305466384776013062133343, −2.53357355407116595555680485350, −1.49931849502870164101229604812, 0.02753599563080778102094712990, 1.64555759343829435786021291020, 2.61339596363219616585378748239, 3.46908288531931645864052440070, 4.57834861727634465686971396363, 5.38960417309694552848297312650, 6.01977586171362523420988742842, 6.70797705603573740790797410188, 7.82508853720076512894804419141, 8.604226577257986711429364283228

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