Properties

Label 4-935712-1.1-c1e2-0-32
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 8·13-s + 16-s − 12·17-s − 10·25-s + 8·26-s − 12·29-s − 32-s + 12·34-s − 8·37-s − 12·41-s + 2·49-s + 10·50-s − 8·52-s − 12·53-s + 12·58-s + 28·61-s + 64-s − 12·68-s + 28·73-s + 8·74-s + 12·82-s + 12·89-s − 20·97-s − 2·98-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2.21·13-s + 1/4·16-s − 2.91·17-s − 2·25-s + 1.56·26-s − 2.22·29-s − 0.176·32-s + 2.05·34-s − 1.31·37-s − 1.87·41-s + 2/7·49-s + 1.41·50-s − 1.10·52-s − 1.64·53-s + 1.57·58-s + 3.58·61-s + 1/8·64-s − 1.45·68-s + 3.27·73-s + 0.929·74-s + 1.32·82-s + 1.27·89-s − 2.03·97-s − 0.202·98-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: $\chi_{935712} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 935712,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−7.72795819597559924442133345234, −7.10678703287277123493279316153, −7.06756557503064388001592546481, −6.59547283064248335138694127148, −6.07507983377375758209992822317, −5.21403289865941834256074312934, −5.21083603966480863402338802043, −4.54790713149621293279441185278, −3.81156610301665935267786910207, −3.58677867797817326821260752741, −2.34092707995523044468378525917, −2.27147129954073815802695201168, −1.81801100280978906223857531826, 0, 0, 1.81801100280978906223857531826, 2.27147129954073815802695201168, 2.34092707995523044468378525917, 3.58677867797817326821260752741, 3.81156610301665935267786910207, 4.54790713149621293279441185278, 5.21083603966480863402338802043, 5.21403289865941834256074312934, 6.07507983377375758209992822317, 6.59547283064248335138694127148, 7.06756557503064388001592546481, 7.10678703287277123493279316153, 7.72795819597559924442133345234

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