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 + 0.685·3-s + 4-s + 1.44·5-s − 0.685·6-s − 7-s − 8-s − 2.52·9-s − 1.44·10-s − 0.0308·11-s + 0.685·12-s + 6.22·13-s + 14-s + 0.994·15-s + 16-s − 5.48·17-s + 2.52·18-s + 1.99·19-s + 1.44·20-s − 0.685·21-s + 0.0308·22-s + 8.11·23-s − 0.685·24-s − 2.89·25-s − 6.22·26-s − 3.79·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.396·3-s + 0.5·4-s + 0.648·5-s − 0.280·6-s − 0.377·7-s − 0.353·8-s − 0.843·9-s − 0.458·10-s − 0.00929·11-s + 0.198·12-s + 1.72·13-s + 0.267·14-s + 0.256·15-s + 0.250·16-s − 1.33·17-s + 0.596·18-s + 0.457·19-s + 0.324·20-s − 0.149·21-s + 0.00657·22-s + 1.69·23-s − 0.140·24-s − 0.579·25-s − 1.21·26-s − 0.729·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.744545581$
$L(\frac12)$  $\approx$  $1.744545581$
$L(\frac{3}{2})$   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 \( 1 - 0.685T + 3T^{2} \)
5 \( 1 - 1.44T + 5T^{2} \)
11 \( 1 + 0.0308T + 11T^{2} \)
13 \( 1 - 6.22T + 13T^{2} \)
17 \( 1 + 5.48T + 17T^{2} \)
19 \( 1 - 1.99T + 19T^{2} \)
23 \( 1 - 8.11T + 23T^{2} \)
29 \( 1 - 2.29T + 29T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 - 6.75T + 37T^{2} \)
41 \( 1 - 5.26T + 41T^{2} \)
43 \( 1 + 5.40T + 43T^{2} \)
47 \( 1 + 3.31T + 47T^{2} \)
53 \( 1 + 2.40T + 53T^{2} \)
59 \( 1 - 4.54T + 59T^{2} \)
61 \( 1 + 6.20T + 61T^{2} \)
67 \( 1 + 0.636T + 67T^{2} \)
71 \( 1 - 1.56T + 71T^{2} \)
73 \( 1 + 8.94T + 73T^{2} \)
79 \( 1 + 7.94T + 79T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 - 9.22T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−8.272181485163639823052042274459, −7.51210640185256184493535800212, −6.48556878199008156388432117424, −6.23921751979605754925550496724, −5.43100905867803612291800811859, −4.37556709733612354568496128989, −3.30728646550822821730571701444, −2.75907427495530973673488461839, −1.79587423686765238233304293341, −0.76433237562782662723588425727, 0.76433237562782662723588425727, 1.79587423686765238233304293341, 2.75907427495530973673488461839, 3.30728646550822821730571701444, 4.37556709733612354568496128989, 5.43100905867803612291800811859, 6.23921751979605754925550496724, 6.48556878199008156388432117424, 7.51210640185256184493535800212, 8.272181485163639823052042274459

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