Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $0.704 + 0.709i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.75·3-s − 1.98i·5-s i·7-s + 4.60·9-s + i·11-s + (−2.54 − 2.55i)13-s + 5.46i·15-s − 7.39·17-s + 4.19i·19-s + 2.75i·21-s + 4.61·23-s + 1.07·25-s − 4.42·27-s + 4.07·29-s − 3.60i·31-s + ⋯
L(s)  = 1  − 1.59·3-s − 0.886i·5-s − 0.377i·7-s + 1.53·9-s + 0.301i·11-s + (−0.704 − 0.709i)13-s + 1.41i·15-s − 1.79·17-s + 0.963i·19-s + 0.601i·21-s + 0.963·23-s + 0.214·25-s − 0.851·27-s + 0.755·29-s − 0.647i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $0.704 + 0.709i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 0.704 + 0.709i)$
$L(1)$  $\approx$  $0.7589341226$
$L(\frac12)$  $\approx$  $0.7589341226$
$L(\frac{3}{2})$   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 \( 1 + (2.54 + 2.55i)T \)
good3 \( 1 + 2.75T + 3T^{2} \)
5 \( 1 + 1.98iT - 5T^{2} \)
17 \( 1 + 7.39T + 17T^{2} \)
19 \( 1 - 4.19iT - 19T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 - 9.67iT - 37T^{2} \)
41 \( 1 - 5.43iT - 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 - 0.284iT - 47T^{2} \)
53 \( 1 + 4.46T + 53T^{2} \)
59 \( 1 - 1.96iT - 59T^{2} \)
61 \( 1 + 2.69T + 61T^{2} \)
67 \( 1 + 12.8iT - 67T^{2} \)
71 \( 1 - 8.00iT - 71T^{2} \)
73 \( 1 - 0.542iT - 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 3.32iT - 83T^{2} \)
89 \( 1 - 14.4iT - 89T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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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

−8.307380595000448237163741983019, −7.50282257349179754923266273936, −6.68615625428160022587882227130, −6.15138317766376104052429261194, −5.25453907472986679043777793838, −4.70054215378723134728691104004, −4.25815694804899556504419280001, −2.76869990674786671753525445031, −1.41262363932963213911395364180, −0.52380652954286586927221298749, 0.58580232054965776565407735544, 2.12375416461109084408972161183, 2.93372325221350235851785838875, 4.33662154117921168679096796162, 4.78647300272984355470456241894, 5.68272654753363561829129196255, 6.36056056117958519326354831317, 7.02305187102446154595774810929, 7.24986541826764479471937054046, 8.846739205427400617503898138938

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