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Properties

Label 2-3024-63.41-c1-0-45
Degree $2$
Conductor $3024$
Sign $-0.973 + 0.228i$
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  + (1.91 − 3.32i)5-s + (0.283 − 2.63i)7-s + (0.585 − 0.338i)11-s + (−4.22 − 2.43i)13-s − 5.79·17-s + 4.22i·19-s + (4.76 + 2.75i)23-s + (−4.86 − 8.41i)25-s + (6.85 − 3.95i)29-s + (−1.78 − 1.02i)31-s + (−8.19 − 5.98i)35-s − 8.71·37-s + (4.84 − 8.38i)41-s + (−3.57 − 6.19i)43-s + (0.666 + 1.15i)47-s + ⋯
L(s)  = 1  + (0.857 − 1.48i)5-s + (0.107 − 0.994i)7-s + (0.176 − 0.101i)11-s + (−1.17 − 0.675i)13-s − 1.40·17-s + 0.969i·19-s + (0.993 + 0.573i)23-s + (−0.972 − 1.68i)25-s + (1.27 − 0.734i)29-s + (−0.320 − 0.184i)31-s + (−1.38 − 1.01i)35-s − 1.43·37-s + (0.755 − 1.30i)41-s + (−0.545 − 0.944i)43-s + (0.0971 + 0.168i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.973 + 0.228i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.973 + 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.412333425\)
\(L(\frac12)\) \(\approx\) \(1.412333425\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.283 + 2.63i)T \)
good5 \( 1 + (-1.91 + 3.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.585 + 0.338i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.22 + 2.43i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.79T + 17T^{2} \)
19 \( 1 - 4.22iT - 19T^{2} \)
23 \( 1 + (-4.76 - 2.75i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.85 + 3.95i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.78 + 1.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.71T + 37T^{2} \)
41 \( 1 + (-4.84 + 8.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.57 + 6.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.666 - 1.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.18iT - 53T^{2} \)
59 \( 1 + (2.09 - 3.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.38 - 1.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.27 + 5.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + 3.65iT - 73T^{2} \)
79 \( 1 + (-5.61 - 9.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.61 + 7.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.79T + 89T^{2} \)
97 \( 1 + (-2.60 + 1.50i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.557777766555409960545789514852, −7.63743078858394690305665823019, −6.93984800198667286414145756202, −5.95931431409590489235279351369, −5.16225192852732283670892900962, −4.63631825852367809595678614865, −3.78770118822178884411700282250, −2.41469520272068219988517564262, −1.44515815536765109990343582214, −0.41252125890039300477559559690, 1.86958526854466192657012479601, 2.57310962401063471067896631907, 3.10118234703563649590095057109, 4.64778452219991354111028986647, 5.14859832114999952784394487980, 6.43053571651317546356296104938, 6.61635815167229313709187486700, 7.26992358517117523248312890774, 8.492117982284972872464994363261, 9.190547251230946175013117467037

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