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Properties

Label 2-1008-252.31-c1-0-14
Degree $2$
Conductor $1008$
Sign $0.993 - 0.110i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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  − 1.73i·3-s + (2 + 1.73i)7-s − 2.99·9-s + 3.46i·11-s + (1.5 + 0.866i)13-s + (1.5 + 0.866i)17-s + (2.5 + 4.33i)19-s + (2.99 − 3.46i)21-s + 3.46i·23-s + 5·25-s + 5.19i·27-s + (−1.5 − 2.59i)29-s + (0.5 + 0.866i)31-s + 5.99·33-s + (−3.5 − 6.06i)37-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.755 + 0.654i)7-s − 0.999·9-s + 1.04i·11-s + (0.416 + 0.240i)13-s + (0.363 + 0.210i)17-s + (0.573 + 0.993i)19-s + (0.654 − 0.755i)21-s + 0.722i·23-s + 25-s + 0.999i·27-s + (−0.278 − 0.482i)29-s + (0.0898 + 0.155i)31-s + 1.04·33-s + (−0.575 − 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.993 - 0.110i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.683051538\)
\(L(\frac12)\) \(\approx\) \(1.683051538\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.5 + 0.866i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.5 + 7.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.873762212411212500194921845436, −9.031842781484416255253837398359, −8.129170924774704354560669787200, −7.58389186831723116870005321559, −6.65894228493631413708576246485, −5.71428089013913827230954363913, −4.96124970429054041813892422460, −3.56885534040834272007776232211, −2.23335594203752738649236048808, −1.39616293828906902073427879420, 0.877432393735979327542944600153, 2.82206866178928575392592792074, 3.69133530146064730787554781732, 4.74467445818543994951173010056, 5.37915883736526696490346205044, 6.47487819326523766144004536533, 7.56668282293507399616367585174, 8.541762962501503292536859148734, 8.978856780261225223881735566595, 10.13926071818423273069872196051

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