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Properties

Label 2-3024-63.4-c1-0-26
Degree $2$
Conductor $3024$
Sign $0.993 - 0.118i$
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  + 4.22·5-s + (2.37 − 1.15i)7-s + 1.92·11-s + (−0.291 + 0.504i)13-s + (−3.61 + 6.25i)17-s + (−2.10 − 3.64i)19-s + 1.27·23-s + 12.8·25-s + (4.20 + 7.27i)29-s + (−0.476 − 0.824i)31-s + (10.0 − 4.89i)35-s + (3.03 + 5.25i)37-s + (−1.31 + 2.27i)41-s + (−0.442 − 0.766i)43-s + (−2.88 + 4.99i)47-s + ⋯
L(s)  = 1  + 1.88·5-s + (0.898 − 0.438i)7-s + 0.581·11-s + (−0.0808 + 0.140i)13-s + (−0.875 + 1.51i)17-s + (−0.482 − 0.835i)19-s + 0.266·23-s + 2.56·25-s + (0.780 + 1.35i)29-s + (−0.0855 − 0.148i)31-s + (1.69 − 0.827i)35-s + (0.498 + 0.863i)37-s + (−0.205 + 0.355i)41-s + (−0.0674 − 0.116i)43-s + (−0.420 + 0.728i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.993 - 0.118i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.993 - 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.128163642\)
\(L(\frac12)\) \(\approx\) \(3.128163642\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.37 + 1.15i)T \)
good5 \( 1 - 4.22T + 5T^{2} \)
11 \( 1 - 1.92T + 11T^{2} \)
13 \( 1 + (0.291 - 0.504i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.61 - 6.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.10 + 3.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.27T + 23T^{2} \)
29 \( 1 + (-4.20 - 7.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.476 + 0.824i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.03 - 5.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.31 - 2.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.442 + 0.766i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.88 - 4.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.962 + 1.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.27 - 3.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.29 + 9.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.43 + 4.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + (-0.446 + 0.772i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.93 - 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.24 + 9.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.87 + 6.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.98 + 3.44i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.711105467141878281004182428997, −8.277955249730072049878370353685, −6.82318635089086214313492247762, −6.62298395372667462151526812775, −5.70151742685366294153945217251, −4.90720573290578171221342547596, −4.23635474010948252447390918902, −2.87352917650746620123653566069, −1.87379058959561504598205405879, −1.31570582458530354409979514829, 1.11745683973974236115995132348, 2.17676567782567423197311694769, 2.60058552100169393578239010116, 4.15997844987681489189228086413, 5.06482868921989302720318883267, 5.59393255497302255112262060785, 6.38265645386399103746097238019, 7.00498199648490158694166270321, 8.116769237210764745056569264474, 8.879303906594009154612335421804

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