Properties

Label 4-199712-1.1-c1e2-0-0
Degree $4$
Conductor $199712$
Sign $1$
Analytic cond. $12.7338$
Root an. cond. $1.88903$
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  − 2-s + 4-s + 6·5-s − 8-s − 5·9-s − 6·10-s + 10·13-s + 16-s + 5·18-s + 6·20-s + 17·25-s − 10·26-s − 32-s − 5·36-s + 4·37-s − 6·40-s − 24·41-s − 30·45-s − 13·49-s − 17·50-s + 10·52-s + 12·53-s + 16·61-s + 64-s + 60·65-s + 5·72-s + 4·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 2.68·5-s − 0.353·8-s − 5/3·9-s − 1.89·10-s + 2.77·13-s + 1/4·16-s + 1.17·18-s + 1.34·20-s + 17/5·25-s − 1.96·26-s − 0.176·32-s − 5/6·36-s + 0.657·37-s − 0.948·40-s − 3.74·41-s − 4.47·45-s − 1.85·49-s − 2.40·50-s + 1.38·52-s + 1.64·53-s + 2.04·61-s + 1/8·64-s + 7.44·65-s + 0.589·72-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(199712\)    =    \(2^{5} \cdot 79^{2}\)
Sign: $1$
Analytic conductor: \(12.7338\)
Root analytic conductor: \(1.88903\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 199712,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.053869146\)
\(L(\frac12)\) \(\approx\) \(2.053869146\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( 1 + T \)
79$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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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

−8.946470375618335270233066681801, −8.713411520077072685722737281970, −8.423485996093001826764876311126, −8.008923035003699396981349994067, −6.88680976179032476598231442371, −6.43698619599801084891178808154, −6.26600987527672149146432415502, −5.66674862285945396423871921055, −5.54299604762027307617963734384, −4.88336349555297367484373677549, −3.47166738807132899057065963424, −3.35403695021279589879198759951, −2.28083472504562987411652929062, −1.90147284566448759002651582081, −1.10580760949883194939213196065, 1.10580760949883194939213196065, 1.90147284566448759002651582081, 2.28083472504562987411652929062, 3.35403695021279589879198759951, 3.47166738807132899057065963424, 4.88336349555297367484373677549, 5.54299604762027307617963734384, 5.66674862285945396423871921055, 6.26600987527672149146432415502, 6.43698619599801084891178808154, 6.88680976179032476598231442371, 8.008923035003699396981349994067, 8.423485996093001826764876311126, 8.713411520077072685722737281970, 8.946470375618335270233066681801

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