↑

Properties

Label 4-2996352-1.1-c1e2-0-7
Degree $4$
Conductor $2996352$
Sign $1$
Analytic cond. $191.050$
Root an. cond. $3.71780$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 16-s + 2·17-s − 8·19-s − 10·25-s + 32-s + 2·34-s − 8·38-s − 12·41-s − 8·43-s − 10·49-s − 10·50-s + 24·59-s + 64-s − 8·67-s + 2·68-s + 4·73-s − 8·76-s − 12·82-s − 24·83-s − 8·86-s + 36·89-s + 28·97-s − 10·98-s − 10·100-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/4·16-s + 0.485·17-s − 1.83·19-s − 2·25-s + 0.176·32-s + 0.342·34-s − 1.29·38-s − 1.87·41-s − 1.21·43-s − 1.42·49-s − 1.41·50-s + 3.12·59-s + 1/8·64-s − 0.977·67-s + 0.242·68-s + 0.468·73-s − 0.917·76-s − 1.32·82-s − 2.63·83-s − 0.862·86-s + 3.81·89-s + 2.84·97-s − 1.01·98-s − 100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2996352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2996352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2996352\)    =    \(2^{7} \cdot 3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(191.050\)
Root analytic conductor: \(3.71780\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2996352,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.352504294\)
\(L(\frac12)\) \(\approx\) \(2.352504294\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 - T \)
3 \( 1 \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.89.abk_ti
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.72163494500314536146419130346, −6.87066981593946766193241441606, −6.70932306148654350305246926067, −6.37184885190800443680649268709, −5.71857773197573233085611833323, −5.60715979666329449780909401068, −4.91664286403250453650685991692, −4.60759445774375135200288872743, −4.09787601221335414896752430182, −3.49564053345359029034543342720, −3.42809860399723990451499242436, −2.54817671292252675229072912013, −1.85853979707166255531246106964, −1.81193945201237649695550582777, −0.48406584317203841332746775292, 0.48406584317203841332746775292, 1.81193945201237649695550582777, 1.85853979707166255531246106964, 2.54817671292252675229072912013, 3.42809860399723990451499242436, 3.49564053345359029034543342720, 4.09787601221335414896752430182, 4.60759445774375135200288872743, 4.91664286403250453650685991692, 5.60715979666329449780909401068, 5.71857773197573233085611833323, 6.37184885190800443680649268709, 6.70932306148654350305246926067, 6.87066981593946766193241441606, 7.72163494500314536146419130346

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