Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $-0.408 + 0.912i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.43·3-s − 3.70i·5-s i·7-s + 2.95·9-s + i·11-s + (1.47 − 3.29i)13-s − 9.03i·15-s + 0.503·17-s + 0.759i·19-s − 2.43i·21-s + 7.83·23-s − 8.70·25-s − 0.114·27-s − 8.07·29-s − 3.66i·31-s + ⋯
L(s)  = 1  + 1.40·3-s − 1.65i·5-s − 0.377i·7-s + 0.984·9-s + 0.301i·11-s + (0.408 − 0.912i)13-s − 2.33i·15-s + 0.122·17-s + 0.174i·19-s − 0.532i·21-s + 1.63·23-s − 1.74·25-s − 0.0219·27-s − 1.50·29-s − 0.657i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.408 + 0.912i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ -0.408 + 0.912i)$
$L(1)$  $\approx$  $3.027134119$
$L(\frac12)$  $\approx$  $3.027134119$
$L(\frac{3}{2})$   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 \( 1 + (-1.47 + 3.29i)T \)
good3 \( 1 - 2.43T + 3T^{2} \)
5 \( 1 + 3.70iT - 5T^{2} \)
17 \( 1 - 0.503T + 17T^{2} \)
19 \( 1 - 0.759iT - 19T^{2} \)
23 \( 1 - 7.83T + 23T^{2} \)
29 \( 1 + 8.07T + 29T^{2} \)
31 \( 1 + 3.66iT - 31T^{2} \)
37 \( 1 + 5.15iT - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + 2.20T + 43T^{2} \)
47 \( 1 - 3.86iT - 47T^{2} \)
53 \( 1 - 7.41T + 53T^{2} \)
59 \( 1 - 3.16iT - 59T^{2} \)
61 \( 1 - 4.30T + 61T^{2} \)
67 \( 1 - 14.7iT - 67T^{2} \)
71 \( 1 + 5.64iT - 71T^{2} \)
73 \( 1 - 14.4iT - 73T^{2} \)
79 \( 1 + 3.53T + 79T^{2} \)
83 \( 1 + 8.51iT - 83T^{2} \)
89 \( 1 - 13.3iT - 89T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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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

−8.443507195216155402615009929364, −7.60795194435808551104648420485, −7.19110994761504050644048206586, −5.68947032349815069384855465658, −5.25322243362815150468226720687, −4.15730428305469291123610317342, −3.71102430265703759334691635400, −2.64039990867189308977323271940, −1.64819333201782656764937585932, −0.69349062186293089139878562972, 1.67307536543779307174350580812, 2.51873383641577025127455993445, 3.23839821561361199027752574243, 3.59764196712166175110742881056, 4.78743047966432161416424834712, 5.93832801924192494129450690051, 6.74537967523022621969848355629, 7.18284417856303150768453982097, 7.985806106402633874619664094418, 8.684411583038151611518208848503

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