Properties

Label 4-1782e2-1.1-c1e2-0-11
Degree $4$
Conductor $3175524$
Sign $1$
Analytic cond. $202.474$
Root an. cond. $3.77217$
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  + 4-s + 3·11-s + 16-s + 12·23-s − 10·25-s − 8·31-s − 8·37-s + 3·44-s + 12·47-s − 10·49-s − 24·53-s − 6·59-s + 64-s + 10·67-s + 24·71-s − 12·89-s + 12·92-s + 10·97-s − 10·100-s + 28·103-s − 12·113-s − 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.904·11-s + 1/4·16-s + 2.50·23-s − 2·25-s − 1.43·31-s − 1.31·37-s + 0.452·44-s + 1.75·47-s − 1.42·49-s − 3.29·53-s − 0.781·59-s + 1/8·64-s + 1.22·67-s + 2.84·71-s − 1.27·89-s + 1.25·92-s + 1.01·97-s − 100-s + 2.75·103-s − 1.12·113-s − 0.181·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3175524\)    =    \(2^{2} \cdot 3^{8} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(202.474\)
Root analytic conductor: \(3.77217\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3175524,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.462983784\)
\(L(\frac12)\) \(\approx\) \(2.462983784\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
11$C_2$ \( 1 - 3 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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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

−7.50395997694989721606241597227, −7.11985565189376005337664653986, −6.68619713403218599725876916726, −6.33800266514324690374051716264, −5.99857165214414943144038732437, −5.34383976963704470462884839275, −5.06687966116029708324180220223, −4.65710913890891472546443799518, −3.83980116686558664178242573817, −3.65076060485342096132488694617, −3.17003897476168898296021928868, −2.57357432097060737204956983168, −1.75073320851877849888240832016, −1.60693251106342657124988453680, −0.58238806619108750440295441942, 0.58238806619108750440295441942, 1.60693251106342657124988453680, 1.75073320851877849888240832016, 2.57357432097060737204956983168, 3.17003897476168898296021928868, 3.65076060485342096132488694617, 3.83980116686558664178242573817, 4.65710913890891472546443799518, 5.06687966116029708324180220223, 5.34383976963704470462884839275, 5.99857165214414943144038732437, 6.33800266514324690374051716264, 6.68619713403218599725876916726, 7.11985565189376005337664653986, 7.50395997694989721606241597227

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