Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 431 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.920·3-s + 4-s + 2.96·5-s + 0.920·6-s − 7-s − 8-s − 2.15·9-s − 2.96·10-s − 3.73·11-s − 0.920·12-s − 4.18·13-s + 14-s − 2.72·15-s + 16-s + 5.64·17-s + 2.15·18-s + 1.59·19-s + 2.96·20-s + 0.920·21-s + 3.73·22-s + 4.99·23-s + 0.920·24-s + 3.79·25-s + 4.18·26-s + 4.74·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.531·3-s + 0.5·4-s + 1.32·5-s + 0.375·6-s − 0.377·7-s − 0.353·8-s − 0.717·9-s − 0.937·10-s − 1.12·11-s − 0.265·12-s − 1.16·13-s + 0.267·14-s − 0.704·15-s + 0.250·16-s + 1.36·17-s + 0.507·18-s + 0.366·19-s + 0.663·20-s + 0.200·21-s + 0.796·22-s + 1.04·23-s + 0.187·24-s + 0.758·25-s + 0.821·26-s + 0.912·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6034\)    =    \(2 \cdot 7 \cdot 431\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9976254970$
$L(\frac12)$  $\approx$  $0.9976254970$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;431\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;431\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 + T \)
431 \( 1 - T \)
good3 \( 1 + 0.920T + 3T^{2} \)
5 \( 1 - 2.96T + 5T^{2} \)
11 \( 1 + 3.73T + 11T^{2} \)
13 \( 1 + 4.18T + 13T^{2} \)
17 \( 1 - 5.64T + 17T^{2} \)
19 \( 1 - 1.59T + 19T^{2} \)
23 \( 1 - 4.99T + 23T^{2} \)
29 \( 1 + 0.281T + 29T^{2} \)
31 \( 1 + 0.595T + 31T^{2} \)
37 \( 1 + 1.75T + 37T^{2} \)
41 \( 1 - 1.43T + 41T^{2} \)
43 \( 1 - 7.07T + 43T^{2} \)
47 \( 1 + 8.92T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 2.59T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 0.827T + 67T^{2} \)
71 \( 1 + 8.62T + 71T^{2} \)
73 \( 1 - 0.950T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 5.53T + 89T^{2} \)
97 \( 1 - 4.82T + 97T^{2} \)
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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

−7.916856999538760020684517529748, −7.52772807667328461064140314847, −6.56376642632769960381576103998, −5.96238419418275659075028993329, −5.30765161367469607354868489208, −4.93303901385811151359528880417, −3.15165801830255993486319895610, −2.74184244657271168229778284440, −1.76931969567070417748719544964, −0.58436541222449823467147406233, 0.58436541222449823467147406233, 1.76931969567070417748719544964, 2.74184244657271168229778284440, 3.15165801830255993486319895610, 4.93303901385811151359528880417, 5.30765161367469607354868489208, 5.96238419418275659075028993329, 6.56376642632769960381576103998, 7.52772807667328461064140314847, 7.916856999538760020684517529748

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