Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11^{2} $
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 − 3-s − 4-s − 2·5-s − 6-s − 7-s − 3·8-s + 9-s − 2·10-s + 12-s − 6·13-s − 14-s + 2·15-s − 16-s − 2·17-s + 18-s − 4·19-s + 2·20-s + 21-s + 3·24-s − 25-s − 6·26-s − 27-s + 28-s + 2·29-s + 2·30-s + 8·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 1.66·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.612·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.365·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2541} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2541,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5960417118$
$L(\frac12)$  $\approx$  $0.5960417118$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.17089028529163, −18.28626651592898, −17.49902436868195, −17.17029994801056, −16.31686154779251, −15.61413183897264, −14.97994505949790, −14.61996134987358, −13.61892639020467, −13.04036046300814, −12.40610205024678, −11.85845612539813, −11.41783230314340, −10.17791660175482, −9.869114401563676, −8.816221995197888, −8.141517002480388, −7.217903736109169, −6.499736857083253, −5.682997860637912, −4.601210256925018, −4.494768071102669, −3.394605363069597, −2.409976073003759, −0.4282831067461068, 0.4282831067461068, 2.409976073003759, 3.394605363069597, 4.494768071102669, 4.601210256925018, 5.682997860637912, 6.499736857083253, 7.217903736109169, 8.141517002480388, 8.816221995197888, 9.869114401563676, 10.17791660175482, 11.41783230314340, 11.85845612539813, 12.40610205024678, 13.04036046300814, 13.61892639020467, 14.61996134987358, 14.97994505949790, 15.61413183897264, 16.31686154779251, 17.17029994801056, 17.49902436868195, 18.28626651592898, 19.17089028529163

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