Properties

Label 2-388416-1.1-c1-0-171
Degree $2$
Conductor $388416$
Sign $-1$
Analytic cond. $3101.51$
Root an. cond. $55.6912$
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 + 3·5-s − 7-s + 9-s + 5·11-s − 3·15-s − 6·19-s + 21-s + 2·23-s + 4·25-s − 27-s − 9·29-s + 3·31-s − 5·33-s − 3·35-s + 6·37-s − 6·41-s + 4·43-s + 3·45-s + 6·47-s + 49-s − 3·53-s + 15·55-s + 6·57-s + 13·59-s + 4·61-s − 63-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.774·15-s − 1.37·19-s + 0.218·21-s + 0.417·23-s + 4/5·25-s − 0.192·27-s − 1.67·29-s + 0.538·31-s − 0.870·33-s − 0.507·35-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 0.447·45-s + 0.875·47-s + 1/7·49-s − 0.412·53-s + 2.02·55-s + 0.794·57-s + 1.69·59-s + 0.512·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388416\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(3101.51\)
Root analytic conductor: \(55.6912\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 388416,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
17 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−12.73133544052532, −12.36189165967891, −11.54940685239981, −11.47122322615912, −10.92887387157810, −10.24891063324725, −10.06103116636497, −9.584165446900076, −9.027202348607313, −8.880558918173784, −8.263563656752662, −7.472975915360837, −6.926399292499115, −6.717013855095925, −6.058438774469668, −5.854250130862974, −5.496944881129889, −4.651838532776674, −4.279547922232526, −3.803434765422880, −3.139130912152984, −2.408793166991948, −1.956794432172486, −1.428993482579253, −0.8533639611465525, 0, 0.8533639611465525, 1.428993482579253, 1.956794432172486, 2.408793166991948, 3.139130912152984, 3.803434765422880, 4.279547922232526, 4.651838532776674, 5.496944881129889, 5.854250130862974, 6.058438774469668, 6.717013855095925, 6.926399292499115, 7.472975915360837, 8.263563656752662, 8.880558918173784, 9.027202348607313, 9.584165446900076, 10.06103116636497, 10.24891063324725, 10.92887387157810, 11.47122322615912, 11.54940685239981, 12.36189165967891, 12.73133544052532

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