Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $0.979 + 0.203i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.10·3-s + 3.39i·5-s i·7-s + 1.44·9-s + i·11-s + (−3.53 − 0.733i)13-s − 7.15i·15-s + 6.05·17-s + 1.08i·19-s + 2.10i·21-s − 6.38·23-s − 6.49·25-s + 3.26·27-s − 1.88·29-s − 5.69i·31-s + ⋯
L(s)  = 1  − 1.21·3-s + 1.51i·5-s − 0.377i·7-s + 0.483·9-s + 0.301i·11-s + (−0.979 − 0.203i)13-s − 1.84i·15-s + 1.46·17-s + 0.248i·19-s + 0.460i·21-s − 1.33·23-s − 1.29·25-s + 0.629·27-s − 0.349·29-s − 1.02i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $0.979 + 0.203i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 0.979 + 0.203i)$
$L(1)$  $\approx$  $0.7471388668$
$L(\frac12)$  $\approx$  $0.7471388668$
$L(\frac{3}{2})$   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 \( 1 + (3.53 + 0.733i)T \)
good3 \( 1 + 2.10T + 3T^{2} \)
5 \( 1 - 3.39iT - 5T^{2} \)
17 \( 1 - 6.05T + 17T^{2} \)
19 \( 1 - 1.08iT - 19T^{2} \)
23 \( 1 + 6.38T + 23T^{2} \)
29 \( 1 + 1.88T + 29T^{2} \)
31 \( 1 + 5.69iT - 31T^{2} \)
37 \( 1 + 5.35iT - 37T^{2} \)
41 \( 1 - 1.69iT - 41T^{2} \)
43 \( 1 - 8.15T + 43T^{2} \)
47 \( 1 + 2.26iT - 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 + 9.70iT - 59T^{2} \)
61 \( 1 - 1.04T + 61T^{2} \)
67 \( 1 - 6.67iT - 67T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 - 8.29iT - 73T^{2} \)
79 \( 1 + 1.23T + 79T^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 + 2.96iT - 89T^{2} \)
97 \( 1 - 0.535iT - 97T^{2} \)
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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

−8.032734686725994307144525246346, −7.52686152108456136098754717414, −6.95157735481327067019796477424, −6.05115127352896601034778121859, −5.75801290463111984551423980351, −4.75578104678587771817836991813, −3.81081849077421552314091758029, −2.96132683623164977822653275070, −1.96307278846603756022735909441, −0.38832585510219955961982776614, 0.71379219390788660961122750014, 1.64090534490189327338291664847, 2.99379945897911601694719969659, 4.25926316478722106031504885132, 4.88217648496367531563984270090, 5.54966669417862155789672979242, 5.88747927801569445700644510659, 6.90823382835388317618175473035, 7.87057975434021886676352765438, 8.398203047086051916730025204531

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