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Properties

Label 2-1008-63.47-c1-0-19
Degree $2$
Conductor $1008$
Sign $0.175 - 0.984i$
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  + (−0.958 + 1.44i)3-s + 4.11·5-s + (2.54 + 0.711i)7-s + (−1.16 − 2.76i)9-s + 5.82i·11-s + (−2.52 + 1.45i)13-s + (−3.94 + 5.93i)15-s + (1.58 + 2.73i)17-s + (−0.722 − 0.417i)19-s + (−3.47 + 2.99i)21-s − 7.09i·23-s + 11.9·25-s + (5.10 + 0.978i)27-s + (−1.91 − 1.10i)29-s + (−3.66 − 2.11i)31-s + ⋯
L(s)  = 1  + (−0.553 + 0.832i)3-s + 1.84·5-s + (0.963 + 0.269i)7-s + (−0.386 − 0.922i)9-s + 1.75i·11-s + (−0.699 + 0.403i)13-s + (−1.01 + 1.53i)15-s + (0.383 + 0.664i)17-s + (−0.165 − 0.0956i)19-s + (−0.757 + 0.653i)21-s − 1.47i·23-s + 2.38·25-s + (0.982 + 0.188i)27-s + (−0.355 − 0.205i)29-s + (−0.658 − 0.380i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.950765626\)
\(L(\frac12)\) \(\approx\) \(1.950765626\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.958 - 1.44i)T \)
7 \( 1 + (-2.54 - 0.711i)T \)
good5 \( 1 - 4.11T + 5T^{2} \)
11 \( 1 - 5.82iT - 11T^{2} \)
13 \( 1 + (2.52 - 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.58 - 2.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.722 + 0.417i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.09iT - 23T^{2} \)
29 \( 1 + (1.91 + 1.10i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.66 + 2.11i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.82 + 3.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.04 - 3.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.155 - 0.269i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.502 + 0.870i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.94 - 1.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.51 - 4.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.98 - 2.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.99 - 8.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (3.04 - 1.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.579 + 1.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.57 + 13.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.82 + 8.35i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.06 - 2.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.30950704793108894453088474278, −9.381642467760356328273915224919, −8.960632581407022596874716817247, −7.55290454703836414000015428960, −6.48859530927700060421735960373, −5.76670685489117116779930494923, −4.87903281780492717557429298958, −4.40441594853334067900579217135, −2.48660310639272322663414410792, −1.70470647402129852454133188197, 1.02744849621368613155535930035, 1.97063189832095599809298118157, 3.10392383227649231297186573528, 5.09042266036615817910160183661, 5.50260538140305065133157392301, 6.17431320673144107718359377976, 7.20513391127534405391268910203, 8.028881553377696211595382953273, 8.963670247494603241347648445634, 9.825239725714027046078784856572

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