Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $0.262 - 0.965i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.18·3-s + 0.656i·5-s i·7-s + 1.76·9-s + i·11-s + (−0.945 + 3.47i)13-s + 1.43i·15-s + 2.46·17-s + 5.98i·19-s − 2.18i·21-s − 1.56·23-s + 4.56·25-s − 2.70·27-s + 1.39·29-s + 7.01i·31-s + ⋯
L(s)  = 1  + 1.25·3-s + 0.293i·5-s − 0.377i·7-s + 0.586·9-s + 0.301i·11-s + (−0.262 + 0.965i)13-s + 0.369i·15-s + 0.596·17-s + 1.37i·19-s − 0.476i·21-s − 0.326·23-s + 0.913·25-s − 0.520·27-s + 0.259·29-s + 1.25i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $0.262 - 0.965i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 0.262 - 0.965i)$
$L(1)$  $\approx$  $2.662375369$
$L(\frac12)$  $\approx$  $2.662375369$
$L(\frac{3}{2})$   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 \( 1 + (0.945 - 3.47i)T \)
good3 \( 1 - 2.18T + 3T^{2} \)
5 \( 1 - 0.656iT - 5T^{2} \)
17 \( 1 - 2.46T + 17T^{2} \)
19 \( 1 - 5.98iT - 19T^{2} \)
23 \( 1 + 1.56T + 23T^{2} \)
29 \( 1 - 1.39T + 29T^{2} \)
31 \( 1 - 7.01iT - 31T^{2} \)
37 \( 1 + 6.51iT - 37T^{2} \)
41 \( 1 - 1.43iT - 41T^{2} \)
43 \( 1 + 2.49T + 43T^{2} \)
47 \( 1 - 2.52iT - 47T^{2} \)
53 \( 1 + 7.93T + 53T^{2} \)
59 \( 1 - 6.65iT - 59T^{2} \)
61 \( 1 - 9.05T + 61T^{2} \)
67 \( 1 - 1.47iT - 67T^{2} \)
71 \( 1 + 4.89iT - 71T^{2} \)
73 \( 1 - 0.638iT - 73T^{2} \)
79 \( 1 + 1.86T + 79T^{2} \)
83 \( 1 - 9.14iT - 83T^{2} \)
89 \( 1 - 8.47iT - 89T^{2} \)
97 \( 1 - 9.88iT - 97T^{2} \)
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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

−8.535787460579331146781263174224, −7.946949365505132129973073210412, −7.26244143643524069873522226427, −6.62947227415885120998979556445, −5.65295759696717488669359934460, −4.63251509800755293368965506465, −3.80608892128824917032743736455, −3.19530755358613899793882509580, −2.25845807826510091440332987777, −1.40301244150890045035112081413, 0.62760279284974971867946858906, 2.00426072064284423414441806018, 2.90192579958384115715345824960, 3.32026229041227983426378541684, 4.47988597599685812470734495099, 5.23295189110049131660212900474, 6.05636204177848766287053471708, 7.03090004525446559208986099682, 7.79960158858807226024789220725, 8.384248383148156660921901846011

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