Properties

Label 4-165564-1.1-c1e2-0-0
Degree $4$
Conductor $165564$
Sign $1$
Analytic cond. $10.5565$
Root an. cond. $1.80251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 3·7-s − 8·13-s + 16-s + 7·19-s + 8·25-s + 3·28-s + 4·31-s − 2·37-s + 43-s − 8·52-s − 5·61-s + 64-s + 28·67-s + 12·73-s + 7·76-s − 14·79-s − 24·91-s + 7·97-s + 8·100-s − 2·103-s − 20·109-s + 3·112-s − 7·121-s + 4·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.13·7-s − 2.21·13-s + 1/4·16-s + 1.60·19-s + 8/5·25-s + 0.566·28-s + 0.718·31-s − 0.328·37-s + 0.152·43-s − 1.10·52-s − 0.640·61-s + 1/8·64-s + 3.42·67-s + 1.40·73-s + 0.802·76-s − 1.57·79-s − 2.51·91-s + 0.710·97-s + 4/5·100-s − 0.197·103-s − 1.91·109-s + 0.283·112-s − 0.636·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(165564\)    =    \(2^{2} \cdot 3^{4} \cdot 7 \cdot 73\)
Sign: $1$
Analytic conductor: \(10.5565\)
Root analytic conductor: \(1.80251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{165564} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 165564,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.076348791\)
\(L(\frac12)\) \(\approx\) \(2.076348791\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
73$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 11 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
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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

−9.315395867058544843961773986566, −8.699528391713256766385124066136, −8.080486028129421915454728815759, −7.79231362179425307530942708107, −7.31136110862664794564680150699, −6.86433893458077974827998245445, −6.47732096389977393293275477001, −5.39352977683038922620478879493, −5.22825377100289241992724167371, −4.82490834230843559592685663185, −4.15813870287003958591964497406, −3.18555535142609381203368985910, −2.67616650614846839500205725771, −2.00623045889002030039972727836, −1.00615658831953362287855636902, 1.00615658831953362287855636902, 2.00623045889002030039972727836, 2.67616650614846839500205725771, 3.18555535142609381203368985910, 4.15813870287003958591964497406, 4.82490834230843559592685663185, 5.22825377100289241992724167371, 5.39352977683038922620478879493, 6.47732096389977393293275477001, 6.86433893458077974827998245445, 7.31136110862664794564680150699, 7.79231362179425307530942708107, 8.080486028129421915454728815759, 8.699528391713256766385124066136, 9.315395867058544843961773986566

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