Properties

Degree 2
Conductor $ 7 \cdot 23 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s − 3-s + 0.618·4-s + 1.23·5-s + 1.61·6-s − 7-s + 2.23·8-s − 2·9-s − 2.00·10-s − 4.47·11-s − 0.618·12-s − 4.23·13-s + 1.61·14-s − 1.23·15-s − 4.85·16-s + 3.23·18-s − 2.76·19-s + 0.763·20-s + 21-s + 7.23·22-s − 23-s − 2.23·24-s − 3.47·25-s + 6.85·26-s + 5·27-s − 0.618·28-s + 7.47·29-s + ⋯
L(s)  = 1  − 1.14·2-s − 0.577·3-s + 0.309·4-s + 0.552·5-s + 0.660·6-s − 0.377·7-s + 0.790·8-s − 0.666·9-s − 0.632·10-s − 1.34·11-s − 0.178·12-s − 1.17·13-s + 0.432·14-s − 0.319·15-s − 1.21·16-s + 0.762·18-s − 0.634·19-s + 0.170·20-s + 0.218·21-s + 1.54·22-s − 0.208·23-s − 0.456·24-s − 0.694·25-s + 1.34·26-s + 0.962·27-s − 0.116·28-s + 1.38·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(161\)    =    \(7 \cdot 23\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{161} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 161,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 + 1.61T + 2T^{2} \)
3 \( 1 + T + 3T^{2} \)
5 \( 1 - 1.23T + 5T^{2} \)
11 \( 1 + 4.47T + 11T^{2} \)
13 \( 1 + 4.23T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 2.76T + 19T^{2} \)
29 \( 1 - 7.47T + 29T^{2} \)
31 \( 1 + 9T + 31T^{2} \)
37 \( 1 - 7.70T + 37T^{2} \)
41 \( 1 - 2.23T + 41T^{2} \)
43 \( 1 + 6.47T + 43T^{2} \)
47 \( 1 - 5.47T + 47T^{2} \)
53 \( 1 - 6.76T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + 5.76T + 71T^{2} \)
73 \( 1 - 6.70T + 73T^{2} \)
79 \( 1 + 2.76T + 79T^{2} \)
83 \( 1 + 2.47T + 83T^{2} \)
89 \( 1 + 8.94T + 89T^{2} \)
97 \( 1 + 9.70T + 97T^{2} \)
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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

−12.28727604748612677389813916111, −10.97500514315244596432963183965, −10.25401701706821133107508669339, −9.458195711122420591134227549652, −8.331537122973244962504931509747, −7.33816763666406712710264269952, −5.94195185101856910183950290577, −4.82721009424870334331669025534, −2.44933117251314136763665246411, 0, 2.44933117251314136763665246411, 4.82721009424870334331669025534, 5.94195185101856910183950290577, 7.33816763666406712710264269952, 8.331537122973244962504931509747, 9.458195711122420591134227549652, 10.25401701706821133107508669339, 10.97500514315244596432963183965, 12.28727604748612677389813916111

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