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Properties

Label 2-88445-1.1-c1-0-47
Degree $2$
Conductor $88445$
Sign $1$
Analytic cond. $706.236$
Root an. cond. $26.5751$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 5-s − 2·9-s − 3·11-s − 2·12-s + 5·13-s + 15-s + 4·16-s − 3·17-s − 2·20-s − 6·23-s + 25-s − 5·27-s − 3·29-s − 4·31-s − 3·33-s + 4·36-s − 2·37-s + 5·39-s − 12·41-s − 10·43-s + 6·44-s − 2·45-s − 9·47-s + 4·48-s − 3·51-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s − 2/3·9-s − 0.904·11-s − 0.577·12-s + 1.38·13-s + 0.258·15-s + 16-s − 0.727·17-s − 0.447·20-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s − 0.718·31-s − 0.522·33-s + 2/3·36-s − 0.328·37-s + 0.800·39-s − 1.87·41-s − 1.52·43-s + 0.904·44-s − 0.298·45-s − 1.31·47-s + 0.577·48-s − 0.420·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88445\)    =    \(5 \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(706.236\)
Root analytic conductor: \(26.5751\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 88445,\ (\ :1/2),\ 1)\)

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)$
bad5 \( 1 - T \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.11570396004970, −13.85028834038035, −13.49441600319200, −13.07288254774849, −12.68266032847331, −11.94591308858119, −11.26071454839350, −10.99860454939520, −10.17650585556805, −9.941649129836045, −9.294601907299570, −8.807204391180863, −8.282540019394144, −8.212673120970553, −7.491106911294140, −6.597379374267446, −6.165274978318877, −5.537045669761310, −5.152111212103307, −4.511242844339877, −3.742778912384551, −3.381944583022466, −2.810593018646920, −1.820156011133551, −1.536657388947387, 0, 0, 1.536657388947387, 1.820156011133551, 2.810593018646920, 3.381944583022466, 3.742778912384551, 4.511242844339877, 5.152111212103307, 5.537045669761310, 6.165274978318877, 6.597379374267446, 7.491106911294140, 8.212673120970553, 8.282540019394144, 8.807204391180863, 9.294601907299570, 9.941649129836045, 10.17650585556805, 10.99860454939520, 11.26071454839350, 11.94591308858119, 12.68266032847331, 13.07288254774849, 13.49441600319200, 13.85028834038035, 14.11570396004970

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