Properties

Label 2-162624-1.1-c1-0-88
Degree $2$
Conductor $162624$
Sign $-1$
Analytic cond. $1298.55$
Root an. cond. $36.0355$
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  + 3-s − 2·5-s − 7-s + 9-s − 6·13-s − 2·15-s + 2·17-s − 8·19-s − 21-s − 4·23-s − 25-s + 27-s + 2·29-s − 8·31-s + 2·35-s − 6·37-s − 6·39-s + 2·41-s + 8·43-s − 2·45-s − 4·47-s + 49-s + 2·51-s − 2·53-s − 8·57-s + 12·59-s + 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s + 0.485·17-s − 1.83·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.986·37-s − 0.960·39-s + 0.312·41-s + 1.21·43-s − 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 1.05·57-s + 1.56·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162624\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1298.55\)
Root analytic conductor: \(36.0355\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{162624} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 162624,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.43506547773995, −12.85988050449513, −12.60628543522407, −12.12409959418050, −11.78044566583290, −11.10394007030713, −10.56752633396718, −10.14819394942863, −9.683781100116631, −9.174388273202851, −8.656656300239655, −8.090086615552739, −7.808751428855513, −7.181264945941810, −6.888725025470485, −6.244187672178562, −5.533605154053738, −5.049820489886196, −4.309724668699712, −3.975012155812640, −3.529426013260171, −2.743206014542686, −2.238805944376964, −1.786588707866377, −0.5861229049956442, 0, 0.5861229049956442, 1.786588707866377, 2.238805944376964, 2.743206014542686, 3.529426013260171, 3.975012155812640, 4.309724668699712, 5.049820489886196, 5.533605154053738, 6.244187672178562, 6.888725025470485, 7.181264945941810, 7.808751428855513, 8.090086615552739, 8.656656300239655, 9.174388273202851, 9.683781100116631, 10.14819394942863, 10.56752633396718, 11.10394007030713, 11.78044566583290, 12.12409959418050, 12.60628543522407, 12.85988050449513, 13.43506547773995

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