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  + (0.900 + 1.56i)2-s + (0.623 − 1.07i)3-s + (−1.12 + 1.94i)4-s + 2.24·6-s − 2.24·8-s + (−0.277 − 0.480i)9-s + (1.40 + 2.42i)12-s + 1.80·13-s + (−0.900 − 1.56i)16-s + (−0.222 + 0.385i)17-s + (0.5 − 0.866i)18-s + (−0.222 − 0.385i)19-s + (−0.623 − 1.07i)23-s + (−1.40 + 2.42i)24-s + (−0.5 + 0.866i)25-s + (1.62 + 2.81i)26-s + ⋯
L(s)  = 1  + (0.900 + 1.56i)2-s + (0.623 − 1.07i)3-s + (−1.12 + 1.94i)4-s + 2.24·6-s − 2.24·8-s + (−0.277 − 0.480i)9-s + (1.40 + 2.42i)12-s + 1.80·13-s + (−0.900 − 1.56i)16-s + (−0.222 + 0.385i)17-s + (0.5 − 0.866i)18-s + (−0.222 − 0.385i)19-s + (−0.623 − 1.07i)23-s + (−1.40 + 2.42i)24-s + (−0.5 + 0.866i)25-s + (1.62 + 2.81i)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  :  $\chi_{2009} (1844, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2009,\ (\ :0),\ 0.0633 - 0.997i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(2.189096510\)
\(L(\frac12)\)  \(\approx\)  \(2.189096510\)
\(L(1)\)   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 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.80T + T^{2} \)
17 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.24T + T^{2} \)
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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

−8.775038965134369957015150194482, −8.416627160696390653759239728244, −7.924379939417520550759319958347, −6.95447597875377976741427818740, −6.49878255232474630489208330425, −5.88521498722584552709073820709, −4.82836953762482163783993782437, −3.90542753497502932814445841013, −3.07571694836446014133026789368, −1.65919010299767152506857707676, 1.39791643264655361206048294006, 2.53557174314562538911207541897, 3.54652072770348459017211618959, 3.87345243392989194229338668329, 4.63375378762089689495879156420, 5.62379910538386516296562473044, 6.35226389030292845089428189104, 7.959681809300946824120904898981, 8.794163293411639487801782865412, 9.501699021753811911276129019053

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