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Properties

Label 2-3024-252.115-c1-0-33
Degree $2$
Conductor $3024$
Sign $0.464 + 0.885i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.10 + 1.21i)5-s + (0.347 − 2.62i)7-s + (−3.88 + 2.24i)11-s + (0.395 − 0.228i)13-s + (1.45 + 0.839i)17-s + (−3.17 − 5.50i)19-s + (−3.03 − 1.75i)23-s + (0.456 + 0.791i)25-s + (1.38 − 2.39i)29-s + 8.92·31-s + (3.92 − 5.10i)35-s + (0.463 + 0.802i)37-s + (9.08 − 5.24i)41-s + (−8.87 − 5.12i)43-s + 8.39·47-s + ⋯
L(s)  = 1  + (0.941 + 0.543i)5-s + (0.131 − 0.991i)7-s + (−1.17 + 0.676i)11-s + (0.109 − 0.0633i)13-s + (0.352 + 0.203i)17-s + (−0.728 − 1.26i)19-s + (−0.633 − 0.365i)23-s + (0.0913 + 0.158i)25-s + (0.256 − 0.444i)29-s + 1.60·31-s + (0.662 − 0.862i)35-s + (0.0762 + 0.132i)37-s + (1.41 − 0.818i)41-s + (−1.35 − 0.781i)43-s + 1.22·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.464 + 0.885i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.464 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.823260954\)
\(L(\frac12)\) \(\approx\) \(1.823260954\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.347 + 2.62i)T \)
good5 \( 1 + (-2.10 - 1.21i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.88 - 2.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.395 + 0.228i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.45 - 0.839i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.17 + 5.50i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.03 + 1.75i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.38 + 2.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.92T + 31T^{2} \)
37 \( 1 + (-0.463 - 0.802i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.08 + 5.24i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.87 + 5.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.39T + 47T^{2} \)
53 \( 1 + (-4.91 + 8.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.45T + 59T^{2} \)
61 \( 1 + 5.31iT - 61T^{2} \)
67 \( 1 + 13.6iT - 67T^{2} \)
71 \( 1 + 1.59iT - 71T^{2} \)
73 \( 1 + (-9.31 - 5.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (-0.657 + 1.13i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.5 - 7.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.69 + 3.28i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.436302350662345099927175967222, −7.86758368349399536787479417671, −6.93395246707830010410667051277, −6.50597373792995577504427845451, −5.52012761153438263139409345121, −4.72982149207254685988642849755, −3.92302957152042537278963016103, −2.64856997340015542390628317055, −2.11726645413096874196934869091, −0.58324382574443631587518119333, 1.22567952617443308939984114432, 2.28442969166119050629401744440, 2.99540929698933495507136605990, 4.28678794476609696449059373111, 5.23367954216346359191906731484, 5.83224621016853990956041310069, 6.18364720889659065591348721480, 7.53056483292198604906544066520, 8.343571453187259284643278919476, 8.657464739050168075653557424999

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