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  − 1.34·2-s − 3-s − 0.202·4-s + 1.16·5-s + 1.34·6-s − 4.05·7-s + 2.95·8-s + 9-s − 1.55·10-s + 0.687·11-s + 0.202·12-s − 5.30·13-s + 5.44·14-s − 1.16·15-s − 3.55·16-s − 17-s − 1.34·18-s + 7.04·19-s − 0.235·20-s + 4.05·21-s − 0.921·22-s + 3.83·23-s − 2.95·24-s − 3.65·25-s + 7.11·26-s − 27-s + 0.822·28-s + ⋯
L(s)  = 1  − 0.947·2-s − 0.577·3-s − 0.101·4-s + 0.519·5-s + 0.547·6-s − 1.53·7-s + 1.04·8-s + 0.333·9-s − 0.492·10-s + 0.207·11-s + 0.0584·12-s − 1.47·13-s + 1.45·14-s − 0.299·15-s − 0.888·16-s − 0.242·17-s − 0.315·18-s + 1.61·19-s − 0.0525·20-s + 0.885·21-s − 0.196·22-s + 0.800·23-s − 0.602·24-s − 0.730·25-s + 1.39·26-s − 0.192·27-s + 0.155·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4058984636$
$L(\frac12)$  $\approx$  $0.4058984636$
$L(\frac{3}{2})$   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 \( 1 + 1.34T + 2T^{2} \)
5 \( 1 - 1.16T + 5T^{2} \)
7 \( 1 + 4.05T + 7T^{2} \)
11 \( 1 - 0.687T + 11T^{2} \)
13 \( 1 + 5.30T + 13T^{2} \)
19 \( 1 - 7.04T + 19T^{2} \)
23 \( 1 - 3.83T + 23T^{2} \)
29 \( 1 - 5.25T + 29T^{2} \)
31 \( 1 + 6.46T + 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 - 7.73T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + 7.75T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 3.00T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 - 4.29T + 71T^{2} \)
73 \( 1 + 2.60T + 73T^{2} \)
83 \( 1 + 5.07T + 83T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 + 7.18T + 97T^{2} \)
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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

−8.700912213638418874642266594646, −7.55014064422467303637497709480, −7.08735311161983826117754157404, −6.45802879683830796434258834082, −5.40148689152506716562497931067, −4.95212928414549576846557702697, −3.76438200411000391314925742666, −2.85388926001337574323807364799, −1.65734772362740292279101676935, −0.43559735900560952820837771987, 0.43559735900560952820837771987, 1.65734772362740292279101676935, 2.85388926001337574323807364799, 3.76438200411000391314925742666, 4.95212928414549576846557702697, 5.40148689152506716562497931067, 6.45802879683830796434258834082, 7.08735311161983826117754157404, 7.55014064422467303637497709480, 8.700912213638418874642266594646

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