Properties

Degree $2$
Conductor $1008$
Sign $-0.927 - 0.373i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.619 − 1.61i)3-s + (−0.590 − 1.02i)5-s + (−0.5 + 0.866i)7-s + (−2.23 − 2.00i)9-s + (−1.85 + 3.20i)11-s + (−0.5 − 0.866i)13-s + (−2.02 + 0.321i)15-s − 6.94·17-s − 1.94·19-s + (1.09 + 1.34i)21-s + (−2.80 − 4.85i)23-s + (1.80 − 3.12i)25-s + (−4.62 + 2.36i)27-s + (−0.119 + 0.207i)29-s + (0.830 + 1.43i)31-s + ⋯
L(s)  = 1  + (0.357 − 0.933i)3-s + (−0.264 − 0.457i)5-s + (−0.188 + 0.327i)7-s + (−0.744 − 0.668i)9-s + (−0.558 + 0.967i)11-s + (−0.138 − 0.240i)13-s + (−0.522 + 0.0830i)15-s − 1.68·17-s − 0.445·19-s + (0.238 + 0.293i)21-s + (−0.584 − 1.01i)23-s + (0.360 − 0.624i)25-s + (−0.890 + 0.455i)27-s + (−0.0222 + 0.0384i)29-s + (0.149 + 0.258i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.927 - 0.373i$
Motivic weight: \(1\)
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3939775941\)
\(L(\frac12)\) \(\approx\) \(0.3939775941\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.619 + 1.61i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.590 + 1.02i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.85 - 3.20i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.94T + 17T^{2} \)
19 \( 1 + 1.94T + 19T^{2} \)
23 \( 1 + (2.80 + 4.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.119 - 0.207i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.830 - 1.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.54T + 37T^{2} \)
41 \( 1 + (-5.09 - 8.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.11 + 1.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-1.30 - 2.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.80 + 6.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.75 - 3.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.60T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + (-3.68 + 6.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.47 - 6.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.74T + 89T^{2} \)
97 \( 1 + (3.58 - 6.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.271941083441159342140071050895, −8.532838890843583645930011234554, −7.965514933090316221943893615943, −6.89175264745603749834164004305, −6.37291993468186173483183333616, −5.08933983818160914651623539277, −4.20494174126303139163850103735, −2.72532738441254203151386483255, −1.93874669782375860681415157286, −0.15615558218268396706815106302, 2.28103040173823964805981785659, 3.34937144359613095821118184064, 4.10224613653032528539750990557, 5.12905990001648044963211064024, 6.11697673985496305540929938962, 7.14479995095904556286821490490, 8.065007100529500665553084387946, 8.890754440887230268925805699500, 9.521892106730941461030068321830, 10.70321559572088518922746320205

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