Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 41^{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 + 6-s − 2·7-s − 8-s + 9-s + 11-s − 12-s + 4·13-s + 2·14-s + 16-s + 6·17-s − 18-s + 4·19-s + 2·21-s − 22-s + 6·23-s + 24-s − 5·25-s − 4·26-s − 27-s − 2·28-s − 6·29-s + 8·31-s − 32-s − 33-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.436·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(110946\)    =    \(2 \cdot 3 \cdot 11 \cdot 41^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{110946} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 110946,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.493817257$
$L(\frac12)$  $\approx$  $1.493817257$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 - T \)
41 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−13.61228142890044, −13.10599994301443, −12.55717975969090, −12.07000511886546, −11.69797829619204, −11.13187388545255, −10.76106167982746, −10.10609540348328, −9.729808520158331, −9.402021504290701, −8.683974237790859, −8.319624573433889, −7.548511626280878, −7.196576573355285, −6.713024261331321, −6.028161166506191, −5.665024848603867, −5.277457213897759, −4.319797324105992, −3.665369348893284, −3.265063348817213, −2.638587634126959, −1.591912176523139, −1.147611206626418, −0.5070697551406463, 0.5070697551406463, 1.147611206626418, 1.591912176523139, 2.638587634126959, 3.265063348817213, 3.665369348893284, 4.319797324105992, 5.277457213897759, 5.665024848603867, 6.028161166506191, 6.713024261331321, 7.196576573355285, 7.548511626280878, 8.319624573433889, 8.683974237790859, 9.402021504290701, 9.729808520158331, 10.10609540348328, 10.76106167982746, 11.13187388545255, 11.69797829619204, 12.07000511886546, 12.55717975969090, 13.10599994301443, 13.61228142890044

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