Properties

Label 4-1782e2-1.1-c1e2-0-21
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 + 6·5-s + 16-s + 6·20-s + 17·25-s − 8·31-s − 2·37-s + 24·47-s + 2·49-s + 12·53-s + 64-s − 8·67-s + 24·71-s + 6·80-s + 6·89-s + 4·97-s + 17·100-s − 8·103-s + 30·113-s − 11·121-s − 8·124-s + 18·125-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + ⋯
L(s)  = 1  + 1/2·4-s + 2.68·5-s + 1/4·16-s + 1.34·20-s + 17/5·25-s − 1.43·31-s − 0.328·37-s + 3.50·47-s + 2/7·49-s + 1.64·53-s + 1/8·64-s − 0.977·67-s + 2.84·71-s + 0.670·80-s + 0.635·89-s + 0.406·97-s + 1.69·100-s − 0.788·103-s + 2.82·113-s − 121-s − 0.718·124-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.164·148-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

\(L(1)\) \(\approx\) \(5.315578076\)
\(L(\frac12)\) \(\approx\) \(5.315578076\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−7.24922326860214571213476798386, −7.24345462277249174674167705434, −6.53027619996145516090571249912, −6.18735538198950166736525445007, −5.89516609795681791126651546534, −5.49090131952205180218546644094, −5.28450148258092108190969844265, −4.77055228819099980126958913880, −3.88992644556742170401529174780, −3.69589098584017718744945020548, −2.76020437849783312932298367516, −2.30016225881827778465495235815, −2.18530917563711206992052355334, −1.51756437179293203913385442880, −0.875045304568839828134702749689, 0.875045304568839828134702749689, 1.51756437179293203913385442880, 2.18530917563711206992052355334, 2.30016225881827778465495235815, 2.76020437849783312932298367516, 3.69589098584017718744945020548, 3.88992644556742170401529174780, 4.77055228819099980126958913880, 5.28450148258092108190969844265, 5.49090131952205180218546644094, 5.89516609795681791126651546534, 6.18735538198950166736525445007, 6.53027619996145516090571249912, 7.24345462277249174674167705434, 7.24922326860214571213476798386

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