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  + 1.31·2-s + 3-s − 0.282·4-s − 0.382·5-s + 1.31·6-s − 2.51·7-s − 2.99·8-s + 9-s − 0.501·10-s − 2.09·11-s − 0.282·12-s + 3.30·13-s − 3.29·14-s − 0.382·15-s − 3.35·16-s + 17-s + 1.31·18-s + 4.13·19-s + 0.108·20-s − 2.51·21-s − 2.74·22-s − 3.29·23-s − 2.99·24-s − 4.85·25-s + 4.33·26-s + 27-s + 0.710·28-s + ⋯
L(s)  = 1  + 0.926·2-s + 0.577·3-s − 0.141·4-s − 0.171·5-s + 0.535·6-s − 0.951·7-s − 1.05·8-s + 0.333·9-s − 0.158·10-s − 0.632·11-s − 0.0814·12-s + 0.917·13-s − 0.881·14-s − 0.0988·15-s − 0.838·16-s + 0.242·17-s + 0.308·18-s + 0.948·19-s + 0.0241·20-s − 0.549·21-s − 0.586·22-s − 0.686·23-s − 0.610·24-s − 0.970·25-s + 0.850·26-s + 0.192·27-s + 0.134·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.648542123$
$L(\frac12)$  $\approx$  $2.648542123$
$L(\frac{3}{2})$   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 \( 1 - 1.31T + 2T^{2} \)
5 \( 1 + 0.382T + 5T^{2} \)
7 \( 1 + 2.51T + 7T^{2} \)
11 \( 1 + 2.09T + 11T^{2} \)
13 \( 1 - 3.30T + 13T^{2} \)
19 \( 1 - 4.13T + 19T^{2} \)
23 \( 1 + 3.29T + 23T^{2} \)
29 \( 1 - 6.91T + 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 - 2.62T + 37T^{2} \)
41 \( 1 + 3.87T + 41T^{2} \)
43 \( 1 + 7.05T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 - 6.71T + 59T^{2} \)
61 \( 1 + 3.61T + 61T^{2} \)
67 \( 1 + 4.26T + 67T^{2} \)
71 \( 1 + 2.45T + 71T^{2} \)
73 \( 1 - 16.3T + 73T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + 2.07T + 89T^{2} \)
97 \( 1 + 9.19T + 97T^{2} \)
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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

−8.378720172308934954497214131088, −7.84793791909780192380845185284, −6.73908796877940433058608057912, −6.15806636459102890717814895315, −5.42158005464490154966169106193, −4.53826288536424092628990862465, −3.75152927607198614008900900490, −3.19093794249563729435881547905, −2.44177047413251887166976859934, −0.78008025842949259207948904758, 0.78008025842949259207948904758, 2.44177047413251887166976859934, 3.19093794249563729435881547905, 3.75152927607198614008900900490, 4.53826288536424092628990862465, 5.42158005464490154966169106193, 6.15806636459102890717814895315, 6.73908796877940433058608057912, 7.84793791909780192380845185284, 8.378720172308934954497214131088

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