Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $0.677 + 0.735i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.334·3-s + 1.98i·5-s i·7-s − 2.88·9-s + i·11-s + (−2.44 − 2.65i)13-s + 0.664i·15-s − 5.84·17-s + 2.86i·19-s − 0.334i·21-s + 0.564·23-s + 1.04·25-s − 1.96·27-s + 8.26·29-s − 1.77i·31-s + ⋯
L(s)  = 1  + 0.192·3-s + 0.889i·5-s − 0.377i·7-s − 0.962·9-s + 0.301i·11-s + (−0.677 − 0.735i)13-s + 0.171i·15-s − 1.41·17-s + 0.658i·19-s − 0.0728i·21-s + 0.117·23-s + 0.208·25-s − 0.378·27-s + 1.53·29-s − 0.317i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $0.677 + 0.735i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 0.677 + 0.735i)$
$L(1)$  $\approx$  $1.258480446$
$L(\frac12)$  $\approx$  $1.258480446$
$L(\frac{3}{2})$   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 \( 1 + (2.44 + 2.65i)T \)
good3 \( 1 - 0.334T + 3T^{2} \)
5 \( 1 - 1.98iT - 5T^{2} \)
17 \( 1 + 5.84T + 17T^{2} \)
19 \( 1 - 2.86iT - 19T^{2} \)
23 \( 1 - 0.564T + 23T^{2} \)
29 \( 1 - 8.26T + 29T^{2} \)
31 \( 1 + 1.77iT - 31T^{2} \)
37 \( 1 - 0.265iT - 37T^{2} \)
41 \( 1 + 5.63iT - 41T^{2} \)
43 \( 1 + 0.932T + 43T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 - 8.63iT - 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 1.98iT - 67T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 + 3.74iT - 73T^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 + 8.15iT - 83T^{2} \)
89 \( 1 + 18.0iT - 89T^{2} \)
97 \( 1 + 12.1iT - 97T^{2} \)
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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

−8.465979132742207240604973580636, −7.49899560745778720760916694669, −6.95938065725382470026267844968, −6.24594276008378044014029258862, −5.40695808180982662479538770748, −4.54641649803523170224788040852, −3.59313032483552382695288950686, −2.77063560131021997204006486134, −2.14735599429111218578474462550, −0.42179918456329904758092609867, 0.903828986076521221328624855859, 2.28752108907896395610581335456, 2.84497253617170422166334295201, 4.12981466477653580758556856580, 4.83739672324836004814496736411, 5.39779663074453622410005247775, 6.46019317579518847489332743987, 6.90908207377607833969319008421, 8.197572849358985555778143010711, 8.492635255366390395037343553644

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