Properties

Degree 2
Conductor $ 3 \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 4·5-s − 2·6-s − 2·7-s + 9-s + 8·10-s − 3·11-s + 2·12-s − 6·13-s + 4·14-s − 4·15-s − 4·16-s + 3·17-s − 2·18-s − 8·20-s − 2·21-s + 6·22-s − 6·23-s + 11·25-s + 12·26-s + 27-s − 4·28-s + 5·29-s + 8·30-s + 7·31-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 1.78·5-s − 0.816·6-s − 0.755·7-s + 1/3·9-s + 2.52·10-s − 0.904·11-s + 0.577·12-s − 1.66·13-s + 1.06·14-s − 1.03·15-s − 16-s + 0.727·17-s − 0.471·18-s − 1.78·20-s − 0.436·21-s + 1.27·22-s − 1.25·23-s + 11/5·25-s + 2.35·26-s + 0.192·27-s − 0.755·28-s + 0.928·29-s + 1.46·30-s + 1.25·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(123\)    =    \(3 \cdot 41\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{123} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 123,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.86405349348129, −19.30182381609248, −18.99965543031521, −17.86737684744474, −16.69218700202976, −15.89958678428256, −15.38488025192687, −14.09611395420063, −12.55068102055940, −11.83698209569362, −10.40776482924956, −9.765611747485775, −8.409047961877698, −7.800913446379695, −7.048559501608216, −4.537455089090865, −2.915454032299703, 0, 2.915454032299703, 4.537455089090865, 7.048559501608216, 7.800913446379695, 8.409047961877698, 9.765611747485775, 10.40776482924956, 11.83698209569362, 12.55068102055940, 14.09611395420063, 15.38488025192687, 15.89958678428256, 16.69218700202976, 17.86737684744474, 18.99965543031521, 19.30182381609248, 19.86405349348129

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