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  − 0.510·2-s − 3-s − 1.73·4-s − 1.60·5-s + 0.510·6-s − 0.772·7-s + 1.90·8-s + 9-s + 0.819·10-s + 5.11·11-s + 1.73·12-s − 4.25·13-s + 0.394·14-s + 1.60·15-s + 2.50·16-s − 17-s − 0.510·18-s + 0.139·19-s + 2.79·20-s + 0.772·21-s − 2.60·22-s + 2.39·23-s − 1.90·24-s − 2.41·25-s + 2.17·26-s − 27-s + 1.34·28-s + ⋯
L(s)  = 1  − 0.360·2-s − 0.577·3-s − 0.869·4-s − 0.718·5-s + 0.208·6-s − 0.292·7-s + 0.674·8-s + 0.333·9-s + 0.259·10-s + 1.54·11-s + 0.502·12-s − 1.18·13-s + 0.105·14-s + 0.414·15-s + 0.626·16-s − 0.242·17-s − 0.120·18-s + 0.0320·19-s + 0.625·20-s + 0.168·21-s − 0.556·22-s + 0.500·23-s − 0.389·24-s − 0.483·25-s + 0.425·26-s − 0.192·27-s + 0.254·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6000540921$
$L(\frac12)$  $\approx$  $0.6000540921$
$L(\frac{3}{2})$   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 \( 1 + 0.510T + 2T^{2} \)
5 \( 1 + 1.60T + 5T^{2} \)
7 \( 1 + 0.772T + 7T^{2} \)
11 \( 1 - 5.11T + 11T^{2} \)
13 \( 1 + 4.25T + 13T^{2} \)
19 \( 1 - 0.139T + 19T^{2} \)
23 \( 1 - 2.39T + 23T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 + 0.727T + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 + 4.55T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 5.80T + 47T^{2} \)
53 \( 1 + 8.80T + 53T^{2} \)
59 \( 1 - 2.77T + 59T^{2} \)
61 \( 1 - 6.83T + 61T^{2} \)
67 \( 1 + 9.93T + 67T^{2} \)
71 \( 1 + 9.38T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
83 \( 1 - 0.155T + 83T^{2} \)
89 \( 1 - 1.13T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−8.462867026245419591490376924808, −7.79286972044543600351268887861, −6.93198905677983721075004693783, −6.44248898496266959299986646344, −5.29668716950589453890064618981, −4.61536720995870003786531205287, −4.03447115381540188670569995698, −3.15653257406791443268559803393, −1.61086614513279614304837053161, −0.50502679549159315109542271185, 0.50502679549159315109542271185, 1.61086614513279614304837053161, 3.15653257406791443268559803393, 4.03447115381540188670569995698, 4.61536720995870003786531205287, 5.29668716950589453890064618981, 6.44248898496266959299986646344, 6.93198905677983721075004693783, 7.79286972044543600351268887861, 8.462867026245419591490376924808

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