Properties

Degree $2$
Conductor $1008$
Sign $-0.0633 + 0.997i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.35 − 1.93i)5-s + (−2.5 + 0.866i)7-s + (−3.35 − 1.93i)11-s − 3.46i·13-s + (−2 − 3.46i)19-s + (6.70 − 3.87i)23-s + (5.00 − 8.66i)25-s − 6.70·29-s + (0.5 − 0.866i)31-s + (−6.70 + 7.74i)35-s + (−2 − 3.46i)37-s + 7.74i·41-s − 6.92i·43-s + (6.70 + 11.6i)47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (1.50 − 0.866i)5-s + (−0.944 + 0.327i)7-s + (−1.01 − 0.583i)11-s − 0.960i·13-s + (−0.458 − 0.794i)19-s + (1.39 − 0.807i)23-s + (1.00 − 1.73i)25-s − 1.24·29-s + (0.0898 − 0.155i)31-s + (−1.13 + 1.30i)35-s + (−0.328 − 0.569i)37-s + 1.20i·41-s − 1.05i·43-s + (0.978 + 1.69i)47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0633 + 0.997i$
Motivic weight: \(1\)
Character: $\chi_{1008} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.531934185\)
\(L(\frac12)\) \(\approx\) \(1.531934185\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (-3.35 + 1.93i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.35 + 1.93i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.70 + 3.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.70T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.74iT - 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 + (-6.70 - 11.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.35 - 5.80i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.35 + 5.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 3.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.74iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + (-6.70 + 3.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−9.649077229409452808444486198422, −9.028295678275931404379353703988, −8.362045797709204464172923510225, −7.11826474052968778207315129160, −6.04379317209646630388971998786, −5.56849098980841387810046279967, −4.75388060518277368634393576174, −3.09178621235459058507616784607, −2.31621911310079668397047597300, −0.66661357118589378072445818320, 1.80282734843396788797130724223, 2.70833146071826184818772484193, 3.77767649679035371814330277991, 5.23117026569129646844917322087, 5.92573974734719527963069184734, 6.91234623730101930454115728588, 7.25118891423320396795971010191, 8.778519205366640803529760236789, 9.587543166703078201905362644870, 10.12162983748689541476447963816

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