Properties

Label 2-133952-1.1-c1-0-54
Degree $2$
Conductor $133952$
Sign $-1$
Analytic cond. $1069.61$
Root an. cond. $32.7049$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 3·9-s + 13-s + 6·17-s − 4·19-s + 23-s − 5·25-s + 2·29-s + 10·31-s − 4·37-s + 10·41-s − 4·43-s − 6·47-s + 49-s + 2·53-s + 2·59-s − 10·61-s − 3·63-s − 16·67-s − 10·71-s − 2·73-s + 8·79-s + 9·81-s + 16·83-s − 8·89-s + 91-s − 4·97-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s + 0.277·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s − 25-s + 0.371·29-s + 1.79·31-s − 0.657·37-s + 1.56·41-s − 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.274·53-s + 0.260·59-s − 1.28·61-s − 0.377·63-s − 1.95·67-s − 1.18·71-s − 0.234·73-s + 0.900·79-s + 81-s + 1.75·83-s − 0.847·89-s + 0.104·91-s − 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133952\)    =    \(2^{6} \cdot 7 \cdot 13 \cdot 23\)
Sign: $-1$
Analytic conductor: \(1069.61\)
Root analytic conductor: \(32.7049\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{133952} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 133952,\ (\ :1/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
13 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.60359128377229, −13.47411384487471, −12.62020202124650, −12.11875424776487, −11.85094079230001, −11.39723845337015, −10.68645690794447, −10.46786303094973, −9.844543041648834, −9.315240141878746, −8.752579609836994, −8.291754761092504, −7.857865114029179, −7.522135369625689, −6.573328919152165, −6.278067776306810, −5.666794230060019, −5.318221271827123, −4.506353158173403, −4.192687289380097, −3.248359374975772, −3.012701658994708, −2.254248170695650, −1.545482373846534, −0.8606815290884784, 0, 0.8606815290884784, 1.545482373846534, 2.254248170695650, 3.012701658994708, 3.248359374975772, 4.192687289380097, 4.506353158173403, 5.318221271827123, 5.666794230060019, 6.278067776306810, 6.573328919152165, 7.522135369625689, 7.857865114029179, 8.291754761092504, 8.752579609836994, 9.315240141878746, 9.844543041648834, 10.46786303094973, 10.68645690794447, 11.39723845337015, 11.85094079230001, 12.11875424776487, 12.62020202124650, 13.47411384487471, 13.60359128377229

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