Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $-0.363 - 0.931i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.97·3-s + 4.37i·5-s i·7-s + 5.83·9-s + i·11-s + (1.31 + 3.35i)13-s + 13.0i·15-s − 5.05·17-s + 0.365i·19-s − 2.97i·21-s − 3.66·23-s − 14.1·25-s + 8.42·27-s + 8.10·29-s − 7.09i·31-s + ⋯
L(s)  = 1  + 1.71·3-s + 1.95i·5-s − 0.377i·7-s + 1.94·9-s + 0.301i·11-s + (0.363 + 0.931i)13-s + 3.35i·15-s − 1.22·17-s + 0.0837i·19-s − 0.648i·21-s − 0.764·23-s − 2.83·25-s + 1.62·27-s + 1.50·29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.363 - 0.931i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ -0.363 - 0.931i)$
$L(1)$  $\approx$  $3.345485232$
$L(\frac12)$  $\approx$  $3.345485232$
$L(\frac{3}{2})$   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 \( 1 + (-1.31 - 3.35i)T \)
good3 \( 1 - 2.97T + 3T^{2} \)
5 \( 1 - 4.37iT - 5T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 - 0.365iT - 19T^{2} \)
23 \( 1 + 3.66T + 23T^{2} \)
29 \( 1 - 8.10T + 29T^{2} \)
31 \( 1 + 7.09iT - 31T^{2} \)
37 \( 1 - 6.97iT - 37T^{2} \)
41 \( 1 - 3.30iT - 41T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 - 8.81T + 53T^{2} \)
59 \( 1 + 8.55iT - 59T^{2} \)
61 \( 1 + 1.88T + 61T^{2} \)
67 \( 1 - 12.4iT - 67T^{2} \)
71 \( 1 - 5.58iT - 71T^{2} \)
73 \( 1 + 1.87iT - 73T^{2} \)
79 \( 1 - 0.669T + 79T^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 - 8.69iT - 89T^{2} \)
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

−8.539642020388468241891807680003, −7.932151796631120002624556624377, −7.25431453377739711151665350864, −6.68379171961044124035731675353, −6.16362878789557687161403012839, −4.30574625554738720463631381136, −4.04560355747587710233071472158, −3.00068654439985516242372384182, −2.52981334937658678264570800402, −1.76760671507219167985140836538, 0.71083394380906275446218785625, 1.79018538501052847042082728509, 2.56184328562251917131965527569, 3.64498234548286335188109567110, 4.28094990469653159433985372525, 5.07058000835337969461411252541, 5.84540508056115911319870681193, 6.98757459471068740167843300031, 7.990704439885088510031553728822, 8.359628539128331121229975857352

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