Properties

Label 2-165-1.1-c5-0-29
Degree $2$
Conductor $165$
Sign $-1$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.17·2-s + 9·3-s − 5.19·4-s + 25·5-s + 46.5·6-s − 123.·7-s − 192.·8-s + 81·9-s + 129.·10-s − 121·11-s − 46.7·12-s − 500.·13-s − 639.·14-s + 225·15-s − 830.·16-s − 422.·17-s + 419.·18-s − 932.·19-s − 129.·20-s − 1.11e3·21-s − 626.·22-s − 1.22e3·23-s − 1.73e3·24-s + 625·25-s − 2.58e3·26-s + 729·27-s + 641.·28-s + ⋯
L(s)  = 1  + 0.915·2-s + 0.577·3-s − 0.162·4-s + 0.447·5-s + 0.528·6-s − 0.952·7-s − 1.06·8-s + 0.333·9-s + 0.409·10-s − 0.301·11-s − 0.0937·12-s − 0.820·13-s − 0.871·14-s + 0.258·15-s − 0.811·16-s − 0.354·17-s + 0.305·18-s − 0.592·19-s − 0.0726·20-s − 0.549·21-s − 0.275·22-s − 0.482·23-s − 0.614·24-s + 0.200·25-s − 0.751·26-s + 0.192·27-s + 0.154·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
5 \( 1 - 25T \)
11 \( 1 + 121T \)
good2 \( 1 - 5.17T + 32T^{2} \)
7 \( 1 + 123.T + 1.68e4T^{2} \)
13 \( 1 + 500.T + 3.71e5T^{2} \)
17 \( 1 + 422.T + 1.41e6T^{2} \)
19 \( 1 + 932.T + 2.47e6T^{2} \)
23 \( 1 + 1.22e3T + 6.43e6T^{2} \)
29 \( 1 + 2.11e3T + 2.05e7T^{2} \)
31 \( 1 + 159.T + 2.86e7T^{2} \)
37 \( 1 - 5.41e3T + 6.93e7T^{2} \)
41 \( 1 + 1.80e4T + 1.15e8T^{2} \)
43 \( 1 - 6.81e3T + 1.47e8T^{2} \)
47 \( 1 + 1.50e4T + 2.29e8T^{2} \)
53 \( 1 - 1.53e4T + 4.18e8T^{2} \)
59 \( 1 - 2.34e4T + 7.14e8T^{2} \)
61 \( 1 - 1.07e4T + 8.44e8T^{2} \)
67 \( 1 - 1.45e4T + 1.35e9T^{2} \)
71 \( 1 + 2.81e4T + 1.80e9T^{2} \)
73 \( 1 + 2.88e4T + 2.07e9T^{2} \)
79 \( 1 + 8.15e3T + 3.07e9T^{2} \)
83 \( 1 + 1.09e5T + 3.93e9T^{2} \)
89 \( 1 - 6.96e4T + 5.58e9T^{2} \)
97 \( 1 - 9.15e4T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.79625474022083768406984573646, −10.18133385577831479801716202034, −9.466133730767755497954006892179, −8.448841756819302632539310932998, −6.94196864888982131119391313733, −5.87617634421056300436678483836, −4.66526059050335581442195082675, −3.46188667418285481120055372018, −2.35840565968180940761034729307, 0, 2.35840565968180940761034729307, 3.46188667418285481120055372018, 4.66526059050335581442195082675, 5.87617634421056300436678483836, 6.94196864888982131119391313733, 8.448841756819302632539310932998, 9.466133730767755497954006892179, 10.18133385577831479801716202034, 11.79625474022083768406984573646

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