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  + 0.680·2-s + 3-s − 1.53·4-s − 1.16·5-s + 0.680·6-s − 3.22·7-s − 2.40·8-s + 9-s − 0.795·10-s − 3.38·11-s − 1.53·12-s − 0.639·13-s − 2.19·14-s − 1.16·15-s + 1.43·16-s + 17-s + 0.680·18-s − 1.61·19-s + 1.79·20-s − 3.22·21-s − 2.30·22-s + 2.39·23-s − 2.40·24-s − 3.63·25-s − 0.435·26-s + 27-s + 4.95·28-s + ⋯
L(s)  = 1  + 0.481·2-s + 0.577·3-s − 0.768·4-s − 0.522·5-s + 0.277·6-s − 1.21·7-s − 0.850·8-s + 0.333·9-s − 0.251·10-s − 1.02·11-s − 0.443·12-s − 0.177·13-s − 0.585·14-s − 0.301·15-s + 0.359·16-s + 0.242·17-s + 0.160·18-s − 0.371·19-s + 0.401·20-s − 0.703·21-s − 0.491·22-s + 0.498·23-s − 0.491·24-s − 0.726·25-s − 0.0853·26-s + 0.192·27-s + 0.936·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.070403633$
$L(\frac12)$  $\approx$  $1.070403633$
$L(\frac{3}{2})$   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 \( 1 - 0.680T + 2T^{2} \)
5 \( 1 + 1.16T + 5T^{2} \)
7 \( 1 + 3.22T + 7T^{2} \)
11 \( 1 + 3.38T + 11T^{2} \)
13 \( 1 + 0.639T + 13T^{2} \)
19 \( 1 + 1.61T + 19T^{2} \)
23 \( 1 - 2.39T + 23T^{2} \)
29 \( 1 + 7.67T + 29T^{2} \)
31 \( 1 - 0.963T + 31T^{2} \)
37 \( 1 + 1.66T + 37T^{2} \)
41 \( 1 - 2.74T + 41T^{2} \)
43 \( 1 - 9.66T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 4.10T + 53T^{2} \)
59 \( 1 + 4.30T + 59T^{2} \)
61 \( 1 - 13.9T + 61T^{2} \)
67 \( 1 + 7.65T + 67T^{2} \)
71 \( 1 - 8.73T + 71T^{2} \)
73 \( 1 - 0.0689T + 73T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 0.235T + 97T^{2} \)
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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

−8.465471282218839536333398740953, −7.70535620619301475956766290239, −7.13303055422674835337291802769, −6.02303516522259619818103260326, −5.48587162693181195229191530635, −4.47621584622541064064859124567, −3.79400434162164498696655110839, −3.16905756958925237673572005304, −2.34512365753360408596704588714, −0.50900657069477315550948831510, 0.50900657069477315550948831510, 2.34512365753360408596704588714, 3.16905756958925237673572005304, 3.79400434162164498696655110839, 4.47621584622541064064859124567, 5.48587162693181195229191530635, 6.02303516522259619818103260326, 7.13303055422674835337291802769, 7.70535620619301475956766290239, 8.465471282218839536333398740953

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