Properties

Degree $2$
Conductor $1008$
Sign $0.787 - 0.616i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.71 − 0.272i)3-s + (−0.119 − 0.207i)5-s + (−0.5 + 0.866i)7-s + (2.85 − 0.931i)9-s + (−2.56 + 4.43i)11-s + (2.44 + 4.23i)13-s + (−0.260 − 0.321i)15-s + 3.70·17-s − 3.66·19-s + (−0.619 + 1.61i)21-s + (3.71 + 6.42i)23-s + (2.47 − 4.28i)25-s + (4.62 − 2.36i)27-s + (−1.73 + 3.00i)29-s + (−0.358 − 0.621i)31-s + ⋯
L(s)  = 1  + (0.987 − 0.157i)3-s + (−0.0534 − 0.0926i)5-s + (−0.188 + 0.327i)7-s + (0.950 − 0.310i)9-s + (−0.772 + 1.33i)11-s + (0.677 + 1.17i)13-s + (−0.0673 − 0.0830i)15-s + 0.898·17-s − 0.839·19-s + (−0.135 + 0.352i)21-s + (0.773 + 1.34i)23-s + (0.494 − 0.856i)25-s + (0.890 − 0.455i)27-s + (−0.321 + 0.557i)29-s + (−0.0644 − 0.111i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.787 - 0.616i$
Motivic weight: \(1\)
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.787 - 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.196369977\)
\(L(\frac12)\) \(\approx\) \(2.196369977\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.272i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.119 + 0.207i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.56 - 4.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.44 - 4.23i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.70T + 17T^{2} \)
19 \( 1 + 3.66T + 19T^{2} \)
23 \( 1 + (-3.71 - 6.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.73 - 3.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.358 + 0.621i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.60T + 37T^{2} \)
41 \( 1 + (2.80 + 4.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.24 + 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.16 + 3.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.942T + 53T^{2} \)
59 \( 1 + (-3.78 - 6.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.75 - 4.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.330 + 0.571i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 3.66T + 73T^{2} \)
79 \( 1 + (3.11 - 5.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.85 + 8.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 + (-8.57 + 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.930199249676000893156041074650, −9.081202660317039716187030127923, −8.574453292577486638937386717073, −7.42168436179365299033962566778, −7.04014187990760149975857624584, −5.77652674658637086027353784973, −4.60779012261765606414926685487, −3.75576054487435265407405397482, −2.56424208696084448468478371086, −1.62105816068170743284440977895, 0.977744371199471365855692125168, 2.81539813650829824982620371699, 3.26166044699315201512058130211, 4.42835323597690597869210036442, 5.58615581834618453136029034650, 6.49864216357453922929306572848, 7.74235781435002959507129470103, 8.148220579321317417429000342434, 8.916652987954882668304751577397, 9.875416412019438901587905933247

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