Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $-0.345 + 0.938i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.189·3-s − 1.01i·5-s i·7-s − 2.96·9-s + i·11-s + (1.24 − 3.38i)13-s + 0.192i·15-s + 3.81·17-s − 0.826i·19-s + 0.189i·21-s + 5.31·23-s + 3.97·25-s + 1.13·27-s − 4.80·29-s − 2.20i·31-s + ⋯
L(s)  = 1  − 0.109·3-s − 0.453i·5-s − 0.377i·7-s − 0.987·9-s + 0.301i·11-s + (0.345 − 0.938i)13-s + 0.0497i·15-s + 0.924·17-s − 0.189i·19-s + 0.0414i·21-s + 1.10·23-s + 0.794·25-s + 0.218·27-s − 0.891·29-s − 0.395i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.345 + 0.938i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ -0.345 + 0.938i)$
$L(1)$  $\approx$  $1.328421119$
$L(\frac12)$  $\approx$  $1.328421119$
$L(\frac{3}{2})$   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 \( 1 + (-1.24 + 3.38i)T \)
good3 \( 1 + 0.189T + 3T^{2} \)
5 \( 1 + 1.01iT - 5T^{2} \)
17 \( 1 - 3.81T + 17T^{2} \)
19 \( 1 + 0.826iT - 19T^{2} \)
23 \( 1 - 5.31T + 23T^{2} \)
29 \( 1 + 4.80T + 29T^{2} \)
31 \( 1 + 2.20iT - 31T^{2} \)
37 \( 1 + 0.485iT - 37T^{2} \)
41 \( 1 - 5.22iT - 41T^{2} \)
43 \( 1 - 1.56T + 43T^{2} \)
47 \( 1 - 6.95iT - 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 + 0.884iT - 59T^{2} \)
61 \( 1 - 1.77T + 61T^{2} \)
67 \( 1 + 14.9iT - 67T^{2} \)
71 \( 1 + 9.34iT - 71T^{2} \)
73 \( 1 + 4.59iT - 73T^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 + 13.0iT - 83T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 - 9.93iT - 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.995914322020854055455006077655, −7.79041907572564175245910262163, −6.69463972117861201420454772454, −5.93051777714741199307478337091, −5.22177821856054269589162607406, −4.60677989867911906139925036493, −3.39359134775671840547006558179, −2.88773941102239181277768957672, −1.46300521469500490256138741030, −0.42960610312187277195427766120, 1.17812623314071826175400585676, 2.42899603613276474293357459854, 3.18372419938035146668609120924, 3.96010785919672898256015125881, 5.19510924444734655603416185670, 5.60411419042203249776353486769, 6.54027974727368567154786739937, 7.06086485424658666327993318645, 8.036580215266084510801721857406, 8.724872340950293363618576301081

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