Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $0.913 - 0.406i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.47·3-s + 1.22i·5-s i·7-s − 0.839·9-s + i·11-s + (−3.29 + 1.46i)13-s + 1.79i·15-s + 5.50·17-s − 4.73i·19-s − 1.47i·21-s + 8.72·23-s + 3.50·25-s − 5.64·27-s − 1.63·29-s − 5.96i·31-s + ⋯
L(s)  = 1  + 0.848·3-s + 0.547i·5-s − 0.377i·7-s − 0.279·9-s + 0.301i·11-s + (−0.913 + 0.406i)13-s + 0.464i·15-s + 1.33·17-s − 1.08i·19-s − 0.320i·21-s + 1.81·23-s + 0.700·25-s − 1.08·27-s − 0.303·29-s − 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $0.913 - 0.406i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 0.913 - 0.406i)$
$L(1)$  $\approx$  $2.507128014$
$L(\frac12)$  $\approx$  $2.507128014$
$L(\frac{3}{2})$   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 \( 1 + (3.29 - 1.46i)T \)
good3 \( 1 - 1.47T + 3T^{2} \)
5 \( 1 - 1.22iT - 5T^{2} \)
17 \( 1 - 5.50T + 17T^{2} \)
19 \( 1 + 4.73iT - 19T^{2} \)
23 \( 1 - 8.72T + 23T^{2} \)
29 \( 1 + 1.63T + 29T^{2} \)
31 \( 1 + 5.96iT - 31T^{2} \)
37 \( 1 - 9.25iT - 37T^{2} \)
41 \( 1 - 6.84iT - 41T^{2} \)
43 \( 1 - 5.93T + 43T^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 - 3.59T + 53T^{2} \)
59 \( 1 + 2.97iT - 59T^{2} \)
61 \( 1 - 3.03T + 61T^{2} \)
67 \( 1 - 12.8iT - 67T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + 1.51iT - 73T^{2} \)
79 \( 1 - 6.58T + 79T^{2} \)
83 \( 1 - 2.25iT - 83T^{2} \)
89 \( 1 - 9.64iT - 89T^{2} \)
97 \( 1 - 5.06iT - 97T^{2} \)
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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

−8.532449792277701257729878951335, −7.66359739957476540472807073807, −7.17516226164361280216349742629, −6.54832867513899563320506753585, −5.37967030537207674463326711986, −4.74786569117867957975162474238, −3.67984629755790989677583848380, −2.90394573837199581761383483688, −2.39129666738016494324837238358, −0.953151821848482594252479102727, 0.814271828540775429184157294836, 2.02024002351553203673777642487, 3.04109066727659828016097550755, 3.46025919455864637195968220741, 4.71679678465806602888202068075, 5.42577896608315042807292995870, 5.97747468984513004876133158844, 7.30818879055443971784399697302, 7.66318136981623706661166318453, 8.526273768420163441477152811850

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