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Properties

Label 2-3024-63.41-c1-0-38
Degree $2$
Conductor $3024$
Sign $-0.441 + 0.897i$
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  + (0.00869 − 0.0150i)5-s + (−2.50 − 0.851i)7-s + (4.13 − 2.38i)11-s + (−1.31 − 0.759i)13-s + 1.91·17-s − 6.45i·19-s + (3.82 + 2.20i)23-s + (2.49 + 4.32i)25-s + (−1.73 + 1.00i)29-s + (−6.51 − 3.76i)31-s + (−0.0346 + 0.0303i)35-s − 6.10·37-s + (−4.67 + 8.10i)41-s + (1.46 + 2.53i)43-s + (−1.45 − 2.52i)47-s + ⋯
L(s)  = 1  + (0.00389 − 0.00673i)5-s + (−0.946 − 0.321i)7-s + (1.24 − 0.719i)11-s + (−0.364 − 0.210i)13-s + 0.463·17-s − 1.48i·19-s + (0.797 + 0.460i)23-s + (0.499 + 0.865i)25-s + (−0.322 + 0.186i)29-s + (−1.17 − 0.676i)31-s + (−0.00585 + 0.00512i)35-s − 1.00·37-s + (−0.730 + 1.26i)41-s + (0.223 + 0.387i)43-s + (−0.212 − 0.368i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.441 + 0.897i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.441 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.147151208\)
\(L(\frac12)\) \(\approx\) \(1.147151208\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 + 0.851i)T \)
good5 \( 1 + (-0.00869 + 0.0150i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.13 + 2.38i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.31 + 0.759i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.91T + 17T^{2} \)
19 \( 1 + 6.45iT - 19T^{2} \)
23 \( 1 + (-3.82 - 2.20i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.73 - 1.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.51 + 3.76i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.10T + 37T^{2} \)
41 \( 1 + (4.67 - 8.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.46 - 2.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.45 + 2.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.94iT - 53T^{2} \)
59 \( 1 + (4.08 - 7.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.484 - 0.279i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.69 + 6.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + 12.8iT - 73T^{2} \)
79 \( 1 + (4.75 + 8.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.67 + 11.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.00T + 89T^{2} \)
97 \( 1 + (-4.09 + 2.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.736121006595522691819124052064, −7.52529356929044378030838026987, −6.97353874567323894431128548693, −6.32177202244228526922458175183, −5.44930244816455605429553289094, −4.58577772415141814709239604481, −3.41747950479761705072618987223, −3.13843047014925433188738530805, −1.57858148977571582254804926927, −0.37520643942762524111942017993, 1.32578487597393293780653033584, 2.39802907483878758272582852531, 3.52011823024460964045026125635, 4.08643381814916516944835029904, 5.22279051635295805169370875524, 5.95800088698157963760362147514, 6.87884560941950141149403569454, 7.15148150888742946478960193204, 8.398497127516302228486726018832, 8.974514190477976732145470714641

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