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Properties

Label 2-3024-252.103-c1-0-25
Degree $2$
Conductor $3024$
Sign $0.859 - 0.511i$
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  + (0.243 − 0.140i)5-s + (2.20 + 1.46i)7-s + (2.44 + 1.41i)11-s + (4.06 + 2.34i)13-s + (7.00 − 4.04i)17-s + (−0.474 + 0.821i)19-s + (0.339 − 0.196i)23-s + (−2.46 + 4.26i)25-s + (−1.51 − 2.61i)29-s − 2.12·31-s + (0.741 + 0.0462i)35-s + (−2.43 + 4.21i)37-s + (−0.478 − 0.276i)41-s + (4.28 − 2.47i)43-s + 2.78·47-s + ⋯
L(s)  = 1  + (0.108 − 0.0627i)5-s + (0.833 + 0.552i)7-s + (0.736 + 0.425i)11-s + (1.12 + 0.651i)13-s + (1.69 − 0.980i)17-s + (−0.108 + 0.188i)19-s + (0.0708 − 0.0409i)23-s + (−0.492 + 0.852i)25-s + (−0.280 − 0.486i)29-s − 0.382·31-s + (0.125 + 0.00781i)35-s + (−0.400 + 0.693i)37-s + (−0.0748 − 0.0431i)41-s + (0.653 − 0.377i)43-s + 0.406·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.467449678\)
\(L(\frac12)\) \(\approx\) \(2.467449678\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 1.46i)T \)
good5 \( 1 + (-0.243 + 0.140i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.44 - 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.06 - 2.34i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-7.00 + 4.04i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.474 - 0.821i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.339 + 0.196i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.51 + 2.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.12T + 31T^{2} \)
37 \( 1 + (2.43 - 4.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.478 + 0.276i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.28 + 2.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.78T + 47T^{2} \)
53 \( 1 + (6.21 + 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 - 9.07iT - 67T^{2} \)
71 \( 1 - 1.54iT - 71T^{2} \)
73 \( 1 + (0.542 - 0.313i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 15.4iT - 79T^{2} \)
83 \( 1 + (-5.30 - 9.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.90 + 5.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.6 + 7.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.842049553794769871526710636456, −8.028520290214385196139409264362, −7.39011891533432291483714339582, −6.47815843317042966807893698749, −5.66127722241342367648563237499, −5.03765518937566814494290128624, −4.04213198327642584515874888103, −3.25641429223459194325798837583, −1.94933153450478710977798074178, −1.20912450610080308912691558351, 0.954481644612782282723532575476, 1.71232532238607406433021811163, 3.23261877956996944729856400308, 3.82789077736443224403646727245, 4.72041668611224666282399838725, 5.90916472927345154048275216620, 6.05025086824963315641950394879, 7.39402762234176214975724437625, 7.85650084697022040235486430197, 8.612949824964690331119315925198

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