Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $-0.325 - 0.945i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.19·3-s − 3.34i·5-s i·7-s − 1.57·9-s + i·11-s + (1.17 + 3.40i)13-s − 3.98i·15-s − 6.04·17-s − 2.04i·19-s − 1.19i·21-s − 2.12·23-s − 6.16·25-s − 5.45·27-s − 6.58·29-s + 11.0i·31-s + ⋯
L(s)  = 1  + 0.688·3-s − 1.49i·5-s − 0.377i·7-s − 0.525·9-s + 0.301i·11-s + (0.325 + 0.945i)13-s − 1.02i·15-s − 1.46·17-s − 0.468i·19-s − 0.260i·21-s − 0.443·23-s − 1.23·25-s − 1.05·27-s − 1.22·29-s + 1.98i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.325 - 0.945i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ -0.325 - 0.945i)$
$L(1)$  $\approx$  $0.4987035110$
$L(\frac12)$  $\approx$  $0.4987035110$
$L(\frac{3}{2})$   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 \( 1 + (-1.17 - 3.40i)T \)
good3 \( 1 - 1.19T + 3T^{2} \)
5 \( 1 + 3.34iT - 5T^{2} \)
17 \( 1 + 6.04T + 17T^{2} \)
19 \( 1 + 2.04iT - 19T^{2} \)
23 \( 1 + 2.12T + 23T^{2} \)
29 \( 1 + 6.58T + 29T^{2} \)
31 \( 1 - 11.0iT - 31T^{2} \)
37 \( 1 - 8.33iT - 37T^{2} \)
41 \( 1 - 9.38iT - 41T^{2} \)
43 \( 1 - 5.02T + 43T^{2} \)
47 \( 1 + 4.19iT - 47T^{2} \)
53 \( 1 - 5.62T + 53T^{2} \)
59 \( 1 - 0.541iT - 59T^{2} \)
61 \( 1 + 2.94T + 61T^{2} \)
67 \( 1 + 3.32iT - 67T^{2} \)
71 \( 1 + 1.30iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 - 5.90T + 79T^{2} \)
83 \( 1 + 2.01iT - 83T^{2} \)
89 \( 1 + 14.4iT - 89T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−8.567086285665843450961645005409, −8.344550926717298338010037230205, −7.25774202519373755912028082571, −6.56073093951938403777410348204, −5.59344276129037015203400417774, −4.69157918039373678318268692624, −4.28959441169296030605885700434, −3.27969384796963750820275516955, −2.14040010570541630535831481432, −1.33572298027041192219998688973, 0.12005620967287443810154578860, 2.28974366764664500279955211999, 2.44640593792819744839159144406, 3.55766966169518493775817320122, 3.99889144885085604413685633290, 5.61618581937212158131753987582, 5.92054826521433595419474893489, 6.80146415757148972655046072251, 7.67586499766795315491435900965, 8.042808246497242619955881205783

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