Properties

Degree 2
Conductor $ 2 \cdot 17 \cdot 41^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 2·6-s + 4·7-s + 8-s + 9-s − 6·11-s + 2·12-s − 2·13-s + 4·14-s + 16-s + 17-s + 18-s + 4·19-s + 8·21-s − 6·22-s + 2·24-s − 5·25-s − 2·26-s − 4·27-s + 4·28-s − 4·31-s + 32-s − 12·33-s + 34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 1.74·21-s − 1.27·22-s + 0.408·24-s − 25-s − 0.392·26-s − 0.769·27-s + 0.755·28-s − 0.718·31-s + 0.176·32-s − 2.08·33-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57154\)    =    \(2 \cdot 17 \cdot 41^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{57154} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 57154,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
17 \( 1 - T \)
41 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.62185559294523, −14.06211778736626, −13.63295205751870, −13.45411002088285, −12.61482313036287, −12.23693262933721, −11.54784819264918, −11.16773467922447, −10.53474564454704, −10.12100730607322, −9.395727575003135, −8.861483320753064, −8.087352211831896, −7.901812406642175, −7.498958101598048, −7.013399999192559, −5.768283901901817, −5.486044816446002, −5.042830943992067, −4.345793334911463, −3.777493064051272, −2.972360943340543, −2.557318483107616, −2.015136684079117, −1.335556577308960, 0, 1.335556577308960, 2.015136684079117, 2.557318483107616, 2.972360943340543, 3.777493064051272, 4.345793334911463, 5.042830943992067, 5.486044816446002, 5.768283901901817, 7.013399999192559, 7.498958101598048, 7.901812406642175, 8.087352211831896, 8.861483320753064, 9.395727575003135, 10.12100730607322, 10.53474564454704, 11.16773467922447, 11.54784819264918, 12.23693262933721, 12.61482313036287, 13.45411002088285, 13.63295205751870, 14.06211778736626, 14.62185559294523

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