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 + 0.632·3-s + 4-s − 0.321·5-s − 0.632·6-s − 7-s − 8-s − 2.59·9-s + 0.321·10-s − 6.13·11-s + 0.632·12-s − 6.17·13-s + 14-s − 0.203·15-s + 16-s − 4.41·17-s + 2.59·18-s − 1.95·19-s − 0.321·20-s − 0.632·21-s + 6.13·22-s − 1.47·23-s − 0.632·24-s − 4.89·25-s + 6.17·26-s − 3.54·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.365·3-s + 0.5·4-s − 0.143·5-s − 0.258·6-s − 0.377·7-s − 0.353·8-s − 0.866·9-s + 0.101·10-s − 1.85·11-s + 0.182·12-s − 1.71·13-s + 0.267·14-s − 0.0525·15-s + 0.250·16-s − 1.07·17-s + 0.612·18-s − 0.448·19-s − 0.0719·20-s − 0.138·21-s + 1.30·22-s − 0.307·23-s − 0.129·24-s − 0.979·25-s + 1.21·26-s − 0.681·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1832648890$
$L(\frac12)$  $\approx$  $0.1832648890$
$L(\frac{3}{2})$   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 \( 1 - 0.632T + 3T^{2} \)
5 \( 1 + 0.321T + 5T^{2} \)
11 \( 1 + 6.13T + 11T^{2} \)
13 \( 1 + 6.17T + 13T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
19 \( 1 + 1.95T + 19T^{2} \)
23 \( 1 + 1.47T + 23T^{2} \)
29 \( 1 - 7.50T + 29T^{2} \)
31 \( 1 - 0.883T + 31T^{2} \)
37 \( 1 + 6.01T + 37T^{2} \)
41 \( 1 - 5.45T + 41T^{2} \)
43 \( 1 + 3.62T + 43T^{2} \)
47 \( 1 - 3.66T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 5.31T + 61T^{2} \)
67 \( 1 - 7.75T + 67T^{2} \)
71 \( 1 - 3.32T + 71T^{2} \)
73 \( 1 + 0.297T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 3.57T + 97T^{2} \)
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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

−8.046854049720612713724907430105, −7.64371479543672954189198837971, −6.82839979780772846681697621750, −6.05573962981102386885501133819, −5.19883494530508354548174502122, −4.57384284922805287963742401889, −3.30153867174670183137930361977, −2.50134643721869866053882101537, −2.20622741365745597997536534460, −0.22015366979512233102814908325, 0.22015366979512233102814908325, 2.20622741365745597997536534460, 2.50134643721869866053882101537, 3.30153867174670183137930361977, 4.57384284922805287963742401889, 5.19883494530508354548174502122, 6.05573962981102386885501133819, 6.82839979780772846681697621750, 7.64371479543672954189198837971, 8.046854049720612713724907430105

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