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  − 2.56·2-s + 3-s + 4.59·4-s + 3.14·5-s − 2.56·6-s − 0.137·7-s − 6.67·8-s + 9-s − 8.08·10-s + 0.886·11-s + 4.59·12-s + 0.153·13-s + 0.353·14-s + 3.14·15-s + 7.95·16-s + 17-s − 2.56·18-s + 6.70·19-s + 14.4·20-s − 0.137·21-s − 2.27·22-s + 7.75·23-s − 6.67·24-s + 4.89·25-s − 0.394·26-s + 27-s − 0.633·28-s + ⋯
L(s)  = 1  − 1.81·2-s + 0.577·3-s + 2.29·4-s + 1.40·5-s − 1.04·6-s − 0.0520·7-s − 2.36·8-s + 0.333·9-s − 2.55·10-s + 0.267·11-s + 1.32·12-s + 0.0425·13-s + 0.0945·14-s + 0.812·15-s + 1.98·16-s + 0.242·17-s − 0.605·18-s + 1.53·19-s + 3.23·20-s − 0.0300·21-s − 0.485·22-s + 1.61·23-s − 1.36·24-s + 0.979·25-s − 0.0773·26-s + 0.192·27-s − 0.119·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.589768663$
$L(\frac12)$  $\approx$  $1.589768663$
$L(\frac{3}{2})$   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 \( 1 + 2.56T + 2T^{2} \)
5 \( 1 - 3.14T + 5T^{2} \)
7 \( 1 + 0.137T + 7T^{2} \)
11 \( 1 - 0.886T + 11T^{2} \)
13 \( 1 - 0.153T + 13T^{2} \)
19 \( 1 - 6.70T + 19T^{2} \)
23 \( 1 - 7.75T + 23T^{2} \)
29 \( 1 - 4.03T + 29T^{2} \)
31 \( 1 + 6.23T + 31T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 - 4.00T + 41T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 + 8.83T + 47T^{2} \)
53 \( 1 + 1.81T + 53T^{2} \)
59 \( 1 - 1.89T + 59T^{2} \)
61 \( 1 + 6.28T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 - 7.83T + 71T^{2} \)
73 \( 1 - 0.0892T + 73T^{2} \)
83 \( 1 - 6.29T + 83T^{2} \)
89 \( 1 + 9.08T + 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−8.669620937627662633666924998637, −7.82562745982423620530687937315, −7.24470972079960230772877527462, −6.51866497729287976692579769804, −5.81747036553677345179291392515, −4.87502371399805868136156670988, −3.23478615741325507696602043171, −2.63428327160389973401865661945, −1.63358915539955602320298990658, −1.01594040535453018298561865204, 1.01594040535453018298561865204, 1.63358915539955602320298990658, 2.63428327160389973401865661945, 3.23478615741325507696602043171, 4.87502371399805868136156670988, 5.81747036553677345179291392515, 6.51866497729287976692579769804, 7.24470972079960230772877527462, 7.82562745982423620530687937315, 8.669620937627662633666924998637

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