Properties

Degree 2
Conductor $ 7^{2} \cdot 41 $
Sign $0.0633 + 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.222 − 0.385i)2-s + (−0.900 − 1.56i)3-s + (0.400 + 0.694i)4-s − 0.801·6-s + 0.801·8-s + (−1.12 + 1.94i)9-s + (0.722 − 1.25i)12-s + 0.445·13-s + (−0.222 + 0.385i)16-s + (0.623 + 1.07i)17-s + (0.499 + 0.866i)18-s + (0.623 − 1.07i)19-s + (0.900 − 1.56i)23-s + (−0.722 − 1.25i)24-s + (−0.5 − 0.866i)25-s + (0.0990 − 0.171i)26-s + ⋯
L(s)  = 1  + (0.222 − 0.385i)2-s + (−0.900 − 1.56i)3-s + (0.400 + 0.694i)4-s − 0.801·6-s + 0.801·8-s + (−1.12 + 1.94i)9-s + (0.722 − 1.25i)12-s + 0.445·13-s + (−0.222 + 0.385i)16-s + (0.623 + 1.07i)17-s + (0.499 + 0.866i)18-s + (0.623 − 1.07i)19-s + (0.900 − 1.56i)23-s + (−0.722 − 1.25i)24-s + (−0.5 − 0.866i)25-s + (0.0990 − 0.171i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2009\)    =    \(7^{2} \cdot 41\)
\( \varepsilon \)  =  $0.0633 + 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{2009} (901, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2009,\ (\ :0),\ 0.0633 + 0.997i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.137403808\)
\(L(\frac12)\)  \(\approx\)  \(1.137403808\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;41\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - 0.445T + T^{2} \)
17 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - 1.24T + T^{2} \)
47 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.80T + T^{2} \)
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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

−8.852285579607738733376073028902, −8.109312287390145997948611799640, −7.52770341845093308909208361251, −6.71956607652015870626490659319, −6.26609944412538972041720521459, −5.26922037028828391186609481251, −4.24639969694681809920604621951, −2.96112448191744686442098279712, −2.13557826763703809556282800369, −1.03645408969807067672409411965, 1.28325315793794628384740804361, 3.15988414154023773290477849106, 3.93556652893087677963129953892, 5.04401577427435506798130622400, 5.41406783742914857800685793023, 5.99624923973918533493469440353, 6.99548593462048006892645907170, 7.82040818671948578844971855507, 9.224428101381131020868302177269, 9.601281576853198179698593941138

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