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  − 0.719·2-s + 3-s − 1.48·4-s + 1.82·5-s − 0.719·6-s + 0.414·7-s + 2.50·8-s + 9-s − 1.31·10-s − 2.75·11-s − 1.48·12-s + 0.245·13-s − 0.298·14-s + 1.82·15-s + 1.15·16-s + 17-s − 0.719·18-s + 0.386·19-s − 2.71·20-s + 0.414·21-s + 1.98·22-s + 6.58·23-s + 2.50·24-s − 1.65·25-s − 0.176·26-s + 27-s − 0.614·28-s + ⋯
L(s)  = 1  − 0.508·2-s + 0.577·3-s − 0.740·4-s + 0.818·5-s − 0.293·6-s + 0.156·7-s + 0.886·8-s + 0.333·9-s − 0.416·10-s − 0.832·11-s − 0.427·12-s + 0.0681·13-s − 0.0797·14-s + 0.472·15-s + 0.289·16-s + 0.242·17-s − 0.169·18-s + 0.0886·19-s − 0.606·20-s + 0.0905·21-s + 0.423·22-s + 1.37·23-s + 0.511·24-s − 0.330·25-s − 0.0346·26-s + 0.192·27-s − 0.116·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.767048325$
$L(\frac12)$  $\approx$  $1.767048325$
$L(\frac{3}{2})$   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 \( 1 + 0.719T + 2T^{2} \)
5 \( 1 - 1.82T + 5T^{2} \)
7 \( 1 - 0.414T + 7T^{2} \)
11 \( 1 + 2.75T + 11T^{2} \)
13 \( 1 - 0.245T + 13T^{2} \)
19 \( 1 - 0.386T + 19T^{2} \)
23 \( 1 - 6.58T + 23T^{2} \)
29 \( 1 + 0.775T + 29T^{2} \)
31 \( 1 - 2.70T + 31T^{2} \)
37 \( 1 - 7.95T + 37T^{2} \)
41 \( 1 + 4.65T + 41T^{2} \)
43 \( 1 + 1.77T + 43T^{2} \)
47 \( 1 - 1.23T + 47T^{2} \)
53 \( 1 - 3.82T + 53T^{2} \)
59 \( 1 - 2.36T + 59T^{2} \)
61 \( 1 - 5.98T + 61T^{2} \)
67 \( 1 - 9.15T + 67T^{2} \)
71 \( 1 + 3.23T + 71T^{2} \)
73 \( 1 + 3.86T + 73T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 + 0.214T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−8.476907287149823657082666403651, −7.917004666507085227025489292047, −7.23616666472074169492008437734, −6.25461558526168835587736226467, −5.29226110778074461496593027116, −4.82471655141435838099510779288, −3.78877765740468332612222239804, −2.83586911066810752901695186327, −1.89363001016625889877573993108, −0.829730491556810945611693655345, 0.829730491556810945611693655345, 1.89363001016625889877573993108, 2.83586911066810752901695186327, 3.78877765740468332612222239804, 4.82471655141435838099510779288, 5.29226110778074461496593027116, 6.25461558526168835587736226467, 7.23616666472074169492008437734, 7.917004666507085227025489292047, 8.476907287149823657082666403651

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