Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $-0.995 - 0.0946i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.64·3-s − 4.02i·5-s i·7-s − 0.305·9-s + i·11-s + (3.58 + 0.341i)13-s + 6.61i·15-s + 2.93·17-s − 5.26i·19-s + 1.64i·21-s − 6.16·23-s − 11.2·25-s + 5.42·27-s + 2.51·29-s − 9.67i·31-s + ⋯
L(s)  = 1  − 0.947·3-s − 1.80i·5-s − 0.377i·7-s − 0.101·9-s + 0.301i·11-s + (0.995 + 0.0946i)13-s + 1.70i·15-s + 0.712·17-s − 1.20i·19-s + 0.358i·21-s − 1.28·23-s − 2.24·25-s + 1.04·27-s + 0.466·29-s − 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0946i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0946i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.995 - 0.0946i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ -0.995 - 0.0946i)$
$L(1)$  $\approx$  $0.9007564324$
$L(\frac12)$  $\approx$  $0.9007564324$
$L(\frac{3}{2})$   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 \( 1 + (-3.58 - 0.341i)T \)
good3 \( 1 + 1.64T + 3T^{2} \)
5 \( 1 + 4.02iT - 5T^{2} \)
17 \( 1 - 2.93T + 17T^{2} \)
19 \( 1 + 5.26iT - 19T^{2} \)
23 \( 1 + 6.16T + 23T^{2} \)
29 \( 1 - 2.51T + 29T^{2} \)
31 \( 1 + 9.67iT - 31T^{2} \)
37 \( 1 + 0.752iT - 37T^{2} \)
41 \( 1 + 2.05iT - 41T^{2} \)
43 \( 1 - 3.54T + 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 - 7.99T + 53T^{2} \)
59 \( 1 + 3.00iT - 59T^{2} \)
61 \( 1 - 6.61T + 61T^{2} \)
67 \( 1 + 8.37iT - 67T^{2} \)
71 \( 1 - 7.73iT - 71T^{2} \)
73 \( 1 + 6.96iT - 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + 13.4iT - 83T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−8.239976258713757936563210994257, −7.38865176838995852800226020691, −6.35086498873835360046924605170, −5.71295393088186642066257999114, −5.16714432439848910795327597226, −4.38883600142865350344421392034, −3.77550870558990265663274465944, −2.19575882581420532396710855102, −1.01849011405082245746873676770, −0.36420192333410913741039371169, 1.38076473464356170589043462608, 2.68449316703855279296133779130, 3.32732225273110444964077994261, 4.14606481170001080999200382577, 5.55114994670910690703219803136, 5.89947704749484585291561542712, 6.44811639785329015407816300637, 7.16878310272426082929315855246, 8.062548513039465716118062044434, 8.635607916199699449946503567084

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