Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
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

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.927 + 0.373i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -0.927 + 0.373i)\)
\(L(1)\)  \(\approx\)  \(0.3939775941\)
\(L(\frac12)\)  \(\approx\)  \(0.3939775941\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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