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  + 2.62·2-s − 3-s + 4.88·4-s − 3.40·5-s − 2.62·6-s + 4.23·7-s + 7.55·8-s + 9-s − 8.92·10-s − 1.96·11-s − 4.88·12-s + 5.61·13-s + 11.1·14-s + 3.40·15-s + 10.0·16-s − 17-s + 2.62·18-s + 2.79·19-s − 16.6·20-s − 4.23·21-s − 5.14·22-s + 1.24·23-s − 7.55·24-s + 6.57·25-s + 14.7·26-s − 27-s + 20.6·28-s + ⋯
L(s)  = 1  + 1.85·2-s − 0.577·3-s + 2.44·4-s − 1.52·5-s − 1.07·6-s + 1.60·7-s + 2.67·8-s + 0.333·9-s − 2.82·10-s − 0.591·11-s − 1.40·12-s + 1.55·13-s + 2.97·14-s + 0.878·15-s + 2.51·16-s − 0.242·17-s + 0.618·18-s + 0.642·19-s − 3.71·20-s − 0.925·21-s − 1.09·22-s + 0.258·23-s − 1.54·24-s + 1.31·25-s + 2.88·26-s − 0.192·27-s + 3.91·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.220075687$
$L(\frac12)$  $\approx$  $5.220075687$
$L(\frac{3}{2})$   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 \( 1 - 2.62T + 2T^{2} \)
5 \( 1 + 3.40T + 5T^{2} \)
7 \( 1 - 4.23T + 7T^{2} \)
11 \( 1 + 1.96T + 11T^{2} \)
13 \( 1 - 5.61T + 13T^{2} \)
19 \( 1 - 2.79T + 19T^{2} \)
23 \( 1 - 1.24T + 23T^{2} \)
29 \( 1 + 6.00T + 29T^{2} \)
31 \( 1 + 4.35T + 31T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 - 5.16T + 41T^{2} \)
43 \( 1 + 2.71T + 43T^{2} \)
47 \( 1 - 7.49T + 47T^{2} \)
53 \( 1 + 7.14T + 53T^{2} \)
59 \( 1 - 8.86T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 8.73T + 67T^{2} \)
71 \( 1 + 7.32T + 71T^{2} \)
73 \( 1 - 0.609T + 73T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 6.85T + 89T^{2} \)
97 \( 1 - 1.54T + 97T^{2} \)
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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

−7.965941647398201045914971007907, −7.59376359372308184462632620679, −6.88055957011023309313820172743, −5.86547002168266889770568255451, −5.29128030079289544230997196644, −4.65572280851282978369233315685, −3.94726493905623040600123085051, −3.51151194258735887148538010970, −2.22348884796454162227718363267, −1.09984264471865849893000206944, 1.09984264471865849893000206944, 2.22348884796454162227718363267, 3.51151194258735887148538010970, 3.94726493905623040600123085051, 4.65572280851282978369233315685, 5.29128030079289544230997196644, 5.86547002168266889770568255451, 6.88055957011023309313820172743, 7.59376359372308184462632620679, 7.965941647398201045914971007907

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