Properties

Label 4-935712-1.1-c1e2-0-5
Degree $4$
Conductor $935712$
Sign $1$
Analytic cond. $59.6618$
Root an. cond. $2.77922$
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 − 4·5-s − 8-s + 4·10-s + 4·13-s + 16-s + 12·17-s − 4·20-s + 2·25-s − 4·26-s + 4·29-s − 32-s − 12·34-s + 20·37-s + 4·40-s − 20·41-s − 14·49-s − 2·50-s + 4·52-s + 20·53-s − 4·58-s + 28·61-s + 64-s − 16·65-s + 12·68-s − 12·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s + 1.26·10-s + 1.10·13-s + 1/4·16-s + 2.91·17-s − 0.894·20-s + 2/5·25-s − 0.784·26-s + 0.742·29-s − 0.176·32-s − 2.05·34-s + 3.28·37-s + 0.632·40-s − 3.12·41-s − 2·49-s − 0.282·50-s + 0.554·52-s + 2.74·53-s − 0.525·58-s + 3.58·61-s + 1/8·64-s − 1.98·65-s + 1.45·68-s − 1.40·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(935712\)    =    \(2^{5} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(59.6618\)
Root analytic conductor: \(2.77922\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 935712,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.178503068\)
\(L(\frac12)\) \(\approx\) \(1.178503068\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.057224083383321383561004935430, −7.87114970877127535090362209983, −7.54188937600423423101808663065, −6.93381933813641935760483898363, −6.59739773494044819386161896201, −5.88087715977795198376928556594, −5.60441811878021212455097225817, −5.02382748416547556621669886282, −4.27229588770101457625379469521, −3.86301997176052042051145556459, −3.33052988139656335920774502455, −3.17802027729570401593965112053, −2.15606001082909168122252072981, −1.15871174429474862980802238748, −0.69408070740984200675842868720, 0.69408070740984200675842868720, 1.15871174429474862980802238748, 2.15606001082909168122252072981, 3.17802027729570401593965112053, 3.33052988139656335920774502455, 3.86301997176052042051145556459, 4.27229588770101457625379469521, 5.02382748416547556621669886282, 5.60441811878021212455097225817, 5.88087715977795198376928556594, 6.59739773494044819386161896201, 6.93381933813641935760483898363, 7.54188937600423423101808663065, 7.87114970877127535090362209983, 8.057224083383321383561004935430

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