Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $-0.935 - 0.353i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.722·3-s + 0.383i·5-s i·7-s − 2.47·9-s + i·11-s + (3.37 + 1.27i)13-s − 0.277i·15-s − 5.43·17-s + 5.21i·19-s + 0.722i·21-s + 2.81·23-s + 4.85·25-s + 3.95·27-s + 1.99·29-s − 2.19i·31-s + ⋯
L(s)  = 1  − 0.417·3-s + 0.171i·5-s − 0.377i·7-s − 0.825·9-s + 0.301i·11-s + (0.935 + 0.353i)13-s − 0.0716i·15-s − 1.31·17-s + 1.19i·19-s + 0.157i·21-s + 0.586·23-s + 0.970·25-s + 0.762·27-s + 0.371·29-s − 0.393i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.935 - 0.353i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ -0.935 - 0.353i)$
$L(1)$  $\approx$  $0.3649555969$
$L(\frac12)$  $\approx$  $0.3649555969$
$L(\frac{3}{2})$   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 \( 1 + (-3.37 - 1.27i)T \)
good3 \( 1 + 0.722T + 3T^{2} \)
5 \( 1 - 0.383iT - 5T^{2} \)
17 \( 1 + 5.43T + 17T^{2} \)
19 \( 1 - 5.21iT - 19T^{2} \)
23 \( 1 - 2.81T + 23T^{2} \)
29 \( 1 - 1.99T + 29T^{2} \)
31 \( 1 + 2.19iT - 31T^{2} \)
37 \( 1 + 8.80iT - 37T^{2} \)
41 \( 1 - 5.89iT - 41T^{2} \)
43 \( 1 + 9.25T + 43T^{2} \)
47 \( 1 + 8.28iT - 47T^{2} \)
53 \( 1 + 0.632T + 53T^{2} \)
59 \( 1 + 13.2iT - 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 - 7.84iT - 67T^{2} \)
71 \( 1 - 8.46iT - 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 13.5iT - 83T^{2} \)
89 \( 1 + 3.98iT - 89T^{2} \)
97 \( 1 - 4.98iT - 97T^{2} \)
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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

−8.597995051266613570246801705653, −8.255516359732080993290358237266, −7.08964567020634499086289340168, −6.59119506329760776719840156602, −5.88868660466297385265948925081, −5.07467070864666516989202872356, −4.22436154600892869061912213417, −3.42847468574235333079756777229, −2.41328391846593415836975827513, −1.28625940444814624025624080893, 0.11785076940784983380772440067, 1.34426890691813442569978122299, 2.73003856605130049876459418329, 3.23332387270482289596245472481, 4.64242403253109417187194679710, 4.99161945005139957304478631578, 6.08177513365083431511564132984, 6.41960094764201219956634566147, 7.30319263290538897323395706572, 8.436875438087322116113964880652

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