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  + 2.59·2-s + 3-s + 4.75·4-s − 1.90·5-s + 2.59·6-s + 1.92·7-s + 7.17·8-s + 9-s − 4.94·10-s − 0.601·11-s + 4.75·12-s + 3.49·13-s + 5.00·14-s − 1.90·15-s + 9.12·16-s + 17-s + 2.59·18-s − 2.11·19-s − 9.04·20-s + 1.92·21-s − 1.56·22-s + 6.97·23-s + 7.17·24-s − 1.38·25-s + 9.07·26-s + 27-s + 9.15·28-s + ⋯
L(s)  = 1  + 1.83·2-s + 0.577·3-s + 2.37·4-s − 0.850·5-s + 1.06·6-s + 0.727·7-s + 2.53·8-s + 0.333·9-s − 1.56·10-s − 0.181·11-s + 1.37·12-s + 0.968·13-s + 1.33·14-s − 0.490·15-s + 2.28·16-s + 0.242·17-s + 0.612·18-s − 0.485·19-s − 2.02·20-s + 0.419·21-s − 0.333·22-s + 1.45·23-s + 1.46·24-s − 0.276·25-s + 1.78·26-s + 0.192·27-s + 1.72·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.718778418$
$L(\frac12)$  $\approx$  $7.718778418$
$L(\frac{3}{2})$   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 \( 1 - 2.59T + 2T^{2} \)
5 \( 1 + 1.90T + 5T^{2} \)
7 \( 1 - 1.92T + 7T^{2} \)
11 \( 1 + 0.601T + 11T^{2} \)
13 \( 1 - 3.49T + 13T^{2} \)
19 \( 1 + 2.11T + 19T^{2} \)
23 \( 1 - 6.97T + 23T^{2} \)
29 \( 1 + 0.266T + 29T^{2} \)
31 \( 1 - 4.68T + 31T^{2} \)
37 \( 1 + 6.70T + 37T^{2} \)
41 \( 1 - 2.69T + 41T^{2} \)
43 \( 1 + 2.92T + 43T^{2} \)
47 \( 1 + 7.17T + 47T^{2} \)
53 \( 1 - 9.70T + 53T^{2} \)
59 \( 1 - 0.899T + 59T^{2} \)
61 \( 1 - 5.08T + 61T^{2} \)
67 \( 1 + 1.11T + 67T^{2} \)
71 \( 1 - 2.37T + 71T^{2} \)
73 \( 1 + 4.92T + 73T^{2} \)
83 \( 1 + 8.55T + 83T^{2} \)
89 \( 1 - 9.51T + 89T^{2} \)
97 \( 1 - 0.679T + 97T^{2} \)
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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

−8.250571752776102876290493731511, −7.52485177397647149983848150477, −6.87335713402766514592657608769, −6.10547688103068354976059711146, −5.15619410310318740489071663151, −4.62740097035166910109340732873, −3.80474565619337678888881556248, −3.32949706048997908482899943629, −2.39163576316242421477086824206, −1.34716091482095992893725062918, 1.34716091482095992893725062918, 2.39163576316242421477086824206, 3.32949706048997908482899943629, 3.80474565619337678888881556248, 4.62740097035166910109340732873, 5.15619410310318740489071663151, 6.10547688103068354976059711146, 6.87335713402766514592657608769, 7.52485177397647149983848150477, 8.250571752776102876290493731511

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