Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $-0.460 + 0.887i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.10·3-s + 1.71i·5-s + i·7-s + 6.62·9-s i·11-s + (1.66 − 3.19i)13-s − 5.32i·15-s + 5.08·17-s − 7.32i·19-s − 3.10i·21-s − 3.22·23-s + 2.05·25-s − 11.2·27-s − 9.04·29-s − 4.98i·31-s + ⋯
L(s)  = 1  − 1.79·3-s + 0.767i·5-s + 0.377i·7-s + 2.20·9-s − 0.301i·11-s + (0.460 − 0.887i)13-s − 1.37i·15-s + 1.23·17-s − 1.67i·19-s − 0.677i·21-s − 0.671·23-s + 0.411·25-s − 2.16·27-s − 1.67·29-s − 0.895i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.460 + 0.887i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ -0.460 + 0.887i)$
$L(1)$  $\approx$  $0.4267087235$
$L(\frac12)$  $\approx$  $0.4267087235$
$L(\frac{3}{2})$   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 \( 1 + (-1.66 + 3.19i)T \)
good3 \( 1 + 3.10T + 3T^{2} \)
5 \( 1 - 1.71iT - 5T^{2} \)
17 \( 1 - 5.08T + 17T^{2} \)
19 \( 1 + 7.32iT - 19T^{2} \)
23 \( 1 + 3.22T + 23T^{2} \)
29 \( 1 + 9.04T + 29T^{2} \)
31 \( 1 + 4.98iT - 31T^{2} \)
37 \( 1 - 1.36iT - 37T^{2} \)
41 \( 1 - 12.1iT - 41T^{2} \)
43 \( 1 - 8.57T + 43T^{2} \)
47 \( 1 - 0.265iT - 47T^{2} \)
53 \( 1 + 9.73T + 53T^{2} \)
59 \( 1 - 0.610iT - 59T^{2} \)
61 \( 1 + 1.47T + 61T^{2} \)
67 \( 1 + 12.4iT - 67T^{2} \)
71 \( 1 - 1.07iT - 71T^{2} \)
73 \( 1 + 7.37iT - 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 - 1.00iT - 83T^{2} \)
89 \( 1 - 2.76iT - 89T^{2} \)
97 \( 1 - 1.77iT - 97T^{2} \)
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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

−7.84862254540411360922535509230, −7.43165605244355596929177355837, −6.38478148814157689089957707561, −6.09753860216170443896045405880, −5.35845306680430900365347852376, −4.71145615912249747491242611098, −3.61245448081727332439360941150, −2.69777577990437399404718998696, −1.26310319086968361175740294418, −0.19427142694525864921658013783, 1.11899625524065515464609239604, 1.73443158427501728353372276728, 3.75353675759459269331733435525, 4.19247484012090637022264831109, 5.18590090639287971874850181770, 5.66234091551089487526509487014, 6.24722415673141870649355620812, 7.19454391457633526036618019988, 7.68406924954389137068988535749, 8.750377307595152010969001540221

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