Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $0.858 + 0.513i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.38·3-s − 0.456i·5-s i·7-s + 8.42·9-s + i·11-s + (−3.09 − 1.85i)13-s − 1.54i·15-s − 0.655·17-s − 4.48i·19-s − 3.38i·21-s + 2.36·23-s + 4.79·25-s + 18.3·27-s + 3.97·29-s + 6.41i·31-s + ⋯
L(s)  = 1  + 1.95·3-s − 0.203i·5-s − 0.377i·7-s + 2.80·9-s + 0.301i·11-s + (−0.858 − 0.513i)13-s − 0.398i·15-s − 0.159·17-s − 1.02i·19-s − 0.737i·21-s + 0.492·23-s + 0.958·25-s + 3.53·27-s + 0.738·29-s + 1.15i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $0.858 + 0.513i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 0.858 + 0.513i)$
$L(1)$  $\approx$  $4.211326738$
$L(\frac12)$  $\approx$  $4.211326738$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 + iT \)
11 \( 1 - iT \)
13 \( 1 + (3.09 + 1.85i)T \)
good3 \( 1 - 3.38T + 3T^{2} \)
5 \( 1 + 0.456iT - 5T^{2} \)
17 \( 1 + 0.655T + 17T^{2} \)
19 \( 1 + 4.48iT - 19T^{2} \)
23 \( 1 - 2.36T + 23T^{2} \)
29 \( 1 - 3.97T + 29T^{2} \)
31 \( 1 - 6.41iT - 31T^{2} \)
37 \( 1 - 5.80iT - 37T^{2} \)
41 \( 1 + 4.28iT - 41T^{2} \)
43 \( 1 - 1.05T + 43T^{2} \)
47 \( 1 - 2.20iT - 47T^{2} \)
53 \( 1 - 2.63T + 53T^{2} \)
59 \( 1 + 4.87iT - 59T^{2} \)
61 \( 1 + 1.09T + 61T^{2} \)
67 \( 1 + 9.64iT - 67T^{2} \)
71 \( 1 + 5.47iT - 71T^{2} \)
73 \( 1 + 16.3iT - 73T^{2} \)
79 \( 1 - 9.07T + 79T^{2} \)
83 \( 1 + 0.226iT - 83T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−8.405927709212889016441068107341, −7.77976341874563881477067040498, −7.10006903040983991231981349838, −6.62064604221132674645946736362, −4.84067852770106335208982218799, −4.71040204129908382210482342358, −3.47265593102314343217818070903, −2.92428888713643252824575755779, −2.13853274926523779253703182197, −1.02152247898120484206072485515, 1.33364172285785466665295217106, 2.42392283816669464817840810727, 2.78982575551838376435698645843, 3.82642900492639068490260408199, 4.39298066596674353495130566225, 5.45729265058461551801473855389, 6.65065421823470171500205277519, 7.20098503231136077997725455601, 7.959806992910924967872216649123, 8.498221255511452225345632376683

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