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Properties

Label 2-1008-7.5-c2-0-24
Degree $2$
Conductor $1008$
Sign $-0.0552 + 0.998i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.79 − 2.19i)5-s + (−0.5 − 6.98i)7-s + (9.79 + 16.9i)11-s − 6.11i·13-s + (−7.59 + 4.38i)17-s + (26.2 + 15.1i)19-s + (12 − 20.7i)23-s + (−2.89 − 5.00i)25-s − 13.5·29-s + (24.2 − 14.0i)31-s + (−13.4 + 27.6i)35-s + (24.2 − 42.0i)37-s − 7.14i·41-s − 53.7·43-s + (−34.5 − 19.9i)47-s + ⋯
L(s)  = 1  + (−0.759 − 0.438i)5-s + (−0.0714 − 0.997i)7-s + (0.890 + 1.54i)11-s − 0.470i·13-s + (−0.446 + 0.257i)17-s + (1.38 + 0.799i)19-s + (0.521 − 0.903i)23-s + (−0.115 − 0.200i)25-s − 0.468·29-s + (0.783 − 0.452i)31-s + (−0.383 + 0.788i)35-s + (0.656 − 1.13i)37-s − 0.174i·41-s − 1.25·43-s + (−0.736 − 0.424i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0552 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0552 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0552 + 0.998i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ -0.0552 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.372179909\)
\(L(\frac12)\) \(\approx\) \(1.372179909\)
\(L(2)\) 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 \( 1 + (0.5 + 6.98i)T \)
good5 \( 1 + (3.79 + 2.19i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-9.79 - 16.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 6.11iT - 169T^{2} \)
17 \( 1 + (7.59 - 4.38i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-26.2 - 15.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-12 + 20.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 13.5T + 841T^{2} \)
31 \( 1 + (-24.2 + 14.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-24.2 + 42.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 7.14iT - 1.68e3T^{2} \)
43 \( 1 + 53.7T + 1.84e3T^{2} \)
47 \( 1 + (34.5 + 19.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (30.7 + 53.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-66.7 + 38.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-0.373 - 0.215i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (28.4 + 49.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 123.T + 5.04e3T^{2} \)
73 \( 1 + (-31.0 + 17.9i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (26.0 - 45.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 136. iT - 6.88e3T^{2} \)
89 \( 1 + (13.2 + 7.63i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 34.1iT - 9.40e3T^{2} \)
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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

−9.764128711854816735651864801214, −8.659819638187401761466334923815, −7.74438300605996358567150124642, −7.20577940805528168383675782936, −6.29972355619185738213744394444, −4.90062877557249142707825492459, −4.25515697721259470884923313273, −3.43104592234540130118093009974, −1.76117786243893122776115914557, −0.49624370118413521534847605356, 1.19256570192344989416217562942, 2.91863213877648502580657368613, 3.45586267470989173747955627533, 4.76485259869591356252942313060, 5.76903599712589562504288823498, 6.59789158126884429027258486648, 7.44946170557673844529764412373, 8.466543972828983768280867262793, 9.060953311204024780706291072069, 9.802421079535242866289861275055

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