Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $-0.866 + 0.498i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.810·3-s + 3.82i·5-s i·7-s − 2.34·9-s + i·11-s + (3.12 − 1.79i)13-s + 3.09i·15-s + 1.91·17-s − 2.75i·19-s − 0.810i·21-s − 7.88·23-s − 9.63·25-s − 4.32·27-s − 8.14·29-s + 6.00i·31-s + ⋯
L(s)  = 1  + 0.467·3-s + 1.71i·5-s − 0.377i·7-s − 0.781·9-s + 0.301i·11-s + (0.866 − 0.498i)13-s + 0.800i·15-s + 0.463·17-s − 0.631i·19-s − 0.176i·21-s − 1.64·23-s − 1.92·25-s − 0.833·27-s − 1.51·29-s + 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.866 + 0.498i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ -0.866 + 0.498i)$
$L(1)$  $\approx$  $0.2861801964$
$L(\frac12)$  $\approx$  $0.2861801964$
$L(\frac{3}{2})$   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 \( 1 + (-3.12 + 1.79i)T \)
good3 \( 1 - 0.810T + 3T^{2} \)
5 \( 1 - 3.82iT - 5T^{2} \)
17 \( 1 - 1.91T + 17T^{2} \)
19 \( 1 + 2.75iT - 19T^{2} \)
23 \( 1 + 7.88T + 23T^{2} \)
29 \( 1 + 8.14T + 29T^{2} \)
31 \( 1 - 6.00iT - 31T^{2} \)
37 \( 1 - 5.83iT - 37T^{2} \)
41 \( 1 + 5.43iT - 41T^{2} \)
43 \( 1 + 5.59T + 43T^{2} \)
47 \( 1 + 12.1iT - 47T^{2} \)
53 \( 1 + 7.25T + 53T^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 1.53iT - 67T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + 9.21iT - 73T^{2} \)
79 \( 1 + 6.37T + 79T^{2} \)
83 \( 1 - 4.00iT - 83T^{2} \)
89 \( 1 + 8.61iT - 89T^{2} \)
97 \( 1 - 1.03iT - 97T^{2} \)
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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

−8.711509276805356094410651873418, −8.104650014474620529720889212947, −7.36915579538683102498874378775, −6.78373840485538389613542364942, −6.01289306584414816768992063805, −5.35451000128519326402716265118, −3.90049342609648054508202430183, −3.44635458673188638011024678644, −2.70508702080332526116766530984, −1.80314162834353908653081489256, 0.07099745035160964689717049169, 1.42499038297170053352295758856, 2.19533201955950676396598804753, 3.55690200465890695298545746603, 4.07146357829540323590038775638, 5.09065321412139579473117944583, 5.83637050062802321227994986041, 6.17463314453843432547676645211, 7.83541998214833088605500441966, 8.040090481488510289062439284922

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