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  + 1.97·2-s + 3-s + 1.88·4-s + 3.63·5-s + 1.97·6-s + 2.53·7-s − 0.220·8-s + 9-s + 7.15·10-s − 3.57·11-s + 1.88·12-s + 1.93·13-s + 5.00·14-s + 3.63·15-s − 4.21·16-s + 17-s + 1.97·18-s − 1.44·19-s + 6.85·20-s + 2.53·21-s − 7.04·22-s − 1.16·23-s − 0.220·24-s + 8.17·25-s + 3.82·26-s + 27-s + 4.79·28-s + ⋯
L(s)  = 1  + 1.39·2-s + 0.577·3-s + 0.944·4-s + 1.62·5-s + 0.804·6-s + 0.959·7-s − 0.0779·8-s + 0.333·9-s + 2.26·10-s − 1.07·11-s + 0.545·12-s + 0.537·13-s + 1.33·14-s + 0.937·15-s − 1.05·16-s + 0.242·17-s + 0.464·18-s − 0.330·19-s + 1.53·20-s + 0.554·21-s − 1.50·22-s − 0.242·23-s − 0.0450·24-s + 1.63·25-s + 0.749·26-s + 0.192·27-s + 0.906·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.143702051$
$L(\frac12)$  $\approx$  $7.143702051$
$L(\frac{3}{2})$   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 \( 1 - 1.97T + 2T^{2} \)
5 \( 1 - 3.63T + 5T^{2} \)
7 \( 1 - 2.53T + 7T^{2} \)
11 \( 1 + 3.57T + 11T^{2} \)
13 \( 1 - 1.93T + 13T^{2} \)
19 \( 1 + 1.44T + 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 - 0.504T + 29T^{2} \)
31 \( 1 - 8.18T + 31T^{2} \)
37 \( 1 + 0.0403T + 37T^{2} \)
41 \( 1 - 2.99T + 41T^{2} \)
43 \( 1 - 2.64T + 43T^{2} \)
47 \( 1 - 5.20T + 47T^{2} \)
53 \( 1 + 5.46T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + 3.38T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 - 5.87T + 71T^{2} \)
73 \( 1 + 4.87T + 73T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + 0.418T + 89T^{2} \)
97 \( 1 + 8.38T + 97T^{2} \)
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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

−8.437100462863303207050622864755, −7.68684393185876930533225798169, −6.62554785004764914053410230104, −5.99366523707268257585756414109, −5.35518142711115855069601479830, −4.79549112171109718244280249592, −3.98255324453384262143729556082, −2.79386958322090089304901073281, −2.39507110870248785161917140941, −1.40243371542794394709371560835, 1.40243371542794394709371560835, 2.39507110870248785161917140941, 2.79386958322090089304901073281, 3.98255324453384262143729556082, 4.79549112171109718244280249592, 5.35518142711115855069601479830, 5.99366523707268257585756414109, 6.62554785004764914053410230104, 7.68684393185876930533225798169, 8.437100462863303207050622864755

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