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Properties

Label 4-199692-1.1-c1e2-0-6
Degree $4$
Conductor $199692$
Sign $1$
Analytic cond. $12.7325$
Root an. cond. $1.88898$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 8·7-s + 9-s − 12-s + 12·13-s + 16-s − 8·19-s − 8·21-s − 6·25-s − 27-s + 8·28-s − 16·31-s + 36-s + 4·37-s − 12·39-s − 2·43-s − 48-s + 34·49-s + 12·52-s + 8·57-s + 20·61-s + 8·63-s + 64-s + 24·67-s − 12·73-s + 6·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 3.02·7-s + 1/3·9-s − 0.288·12-s + 3.32·13-s + 1/4·16-s − 1.83·19-s − 1.74·21-s − 6/5·25-s − 0.192·27-s + 1.51·28-s − 2.87·31-s + 1/6·36-s + 0.657·37-s − 1.92·39-s − 0.304·43-s − 0.144·48-s + 34/7·49-s + 1.66·52-s + 1.05·57-s + 2.56·61-s + 1.00·63-s + 1/8·64-s + 2.93·67-s − 1.40·73-s + 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199692 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199692 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(199692\)    =    \(2^{2} \cdot 3^{3} \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(12.7325\)
Root analytic conductor: \(1.88898\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 199692,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.584280852\)
\(L(\frac12)\) \(\approx\) \(2.584280852\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
43$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.839256516861246310549158243199, −8.431244106809584233313098529596, −8.073225194162897967644366211249, −8.053074923430498984481272422442, −7.01817007784847403129570171922, −6.81423271072604050272773407016, −5.89875809676899325705751850229, −5.67121691793086248627773881702, −5.34857079660866437419173667226, −4.42301116378591245230571379491, −3.99608303859767674782634823001, −3.73709481518315656789132229596, −2.22767717792336911349172992876, −1.65654682953384645737437951893, −1.31491108910920543726233044250, 1.31491108910920543726233044250, 1.65654682953384645737437951893, 2.22767717792336911349172992876, 3.73709481518315656789132229596, 3.99608303859767674782634823001, 4.42301116378591245230571379491, 5.34857079660866437419173667226, 5.67121691793086248627773881702, 5.89875809676899325705751850229, 6.81423271072604050272773407016, 7.01817007784847403129570171922, 8.053074923430498984481272422442, 8.073225194162897967644366211249, 8.431244106809584233313098529596, 8.839256516861246310549158243199

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