Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 79 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + 3-s + 1.00·4-s + 2.89·5-s − 1.73·6-s + 3.30·7-s + 1.72·8-s + 9-s − 5.01·10-s + 3.25·11-s + 1.00·12-s − 1.42·13-s − 5.72·14-s + 2.89·15-s − 4.99·16-s + 17-s − 1.73·18-s + 2.32·19-s + 2.90·20-s + 3.30·21-s − 5.63·22-s − 8.40·23-s + 1.72·24-s + 3.38·25-s + 2.46·26-s + 27-s + 3.30·28-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.577·3-s + 0.501·4-s + 1.29·5-s − 0.707·6-s + 1.24·7-s + 0.611·8-s + 0.333·9-s − 1.58·10-s + 0.980·11-s + 0.289·12-s − 0.395·13-s − 1.52·14-s + 0.747·15-s − 1.24·16-s + 0.242·17-s − 0.408·18-s + 0.532·19-s + 0.648·20-s + 0.720·21-s − 1.20·22-s − 1.75·23-s + 0.352·24-s + 0.676·25-s + 0.484·26-s + 0.192·27-s + 0.625·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4029\)    =    \(3 \cdot 17 \cdot 79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4029} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.012927359$
$L(\frac12)$  $\approx$  $2.012927359$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;17,\;79\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;79\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
17 \( 1 - T \)
79 \( 1 - T \)
good2 \( 1 + 1.73T + 2T^{2} \)
5 \( 1 - 2.89T + 5T^{2} \)
7 \( 1 - 3.30T + 7T^{2} \)
11 \( 1 - 3.25T + 11T^{2} \)
13 \( 1 + 1.42T + 13T^{2} \)
19 \( 1 - 2.32T + 19T^{2} \)
23 \( 1 + 8.40T + 23T^{2} \)
29 \( 1 + 0.650T + 29T^{2} \)
31 \( 1 + 0.747T + 31T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 + 9.70T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 6.77T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + 7.73T + 59T^{2} \)
61 \( 1 + 2.72T + 61T^{2} \)
67 \( 1 + 2.41T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 5.37T + 73T^{2} \)
83 \( 1 + 4.72T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + 2.34T + 97T^{2} \)
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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

−8.656710441156363992055708625426, −7.79267719132053541285515749837, −7.40680439032911315754710937589, −6.36195290216377980752568149930, −5.57470738482218669565557177896, −4.66094238934701030666799970529, −3.87545238554676857299766430761, −2.35031798799869208435317772717, −1.82448267745375259551119538119, −1.04740337062937138399439037337, 1.04740337062937138399439037337, 1.82448267745375259551119538119, 2.35031798799869208435317772717, 3.87545238554676857299766430761, 4.66094238934701030666799970529, 5.57470738482218669565557177896, 6.36195290216377980752568149930, 7.40680439032911315754710937589, 7.79267719132053541285515749837, 8.656710441156363992055708625426

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