↑

Properties

Label 4-877952-1.1-c1e2-0-5
Degree $4$
Conductor $877952$
Sign $1$
Analytic cond. $55.9789$
Root an. cond. $2.73530$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·9-s − 10·17-s − 19-s − 10·25-s + 14·27-s − 8·31-s − 5·49-s + 20·51-s + 2·57-s − 2·59-s + 28·61-s − 26·67-s − 20·71-s + 18·73-s + 20·75-s + 20·79-s − 4·81-s + 16·93-s − 28·101-s − 12·103-s − 30·107-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s − 2.42·17-s − 0.229·19-s − 2·25-s + 2.69·27-s − 1.43·31-s − 5/7·49-s + 2.80·51-s + 0.264·57-s − 0.260·59-s + 3.58·61-s − 3.17·67-s − 2.37·71-s + 2.10·73-s + 2.30·75-s + 2.25·79-s − 4/9·81-s + 1.65·93-s − 2.78·101-s − 1.18·103-s − 2.90·107-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(877952\)    =    \(2^{7} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(55.9789\)
Root analytic conductor: \(2.73530\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 877952,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
19$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.17.k_ch
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.a_bx
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.59.c_ep
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.61.abc_mg
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.67.ba_lr
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.71.u_ji
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.73.as_it
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.83.a_co
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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

−7.74613152589584229848027600152, −7.18297506166153508271072468025, −6.72598040565036698033673525843, −6.21150502900434643631796858377, −6.13690005259013972771512187418, −5.41676012615338774687665516924, −5.20164006899702178333342847264, −4.65444054474886627728449087886, −3.93346432196685798564362812764, −3.71573461775705014184176995236, −2.56005012019129152265351485884, −2.44767093282979086637766710358, −1.50567723507471670967120404283, 0, 0, 1.50567723507471670967120404283, 2.44767093282979086637766710358, 2.56005012019129152265351485884, 3.71573461775705014184176995236, 3.93346432196685798564362812764, 4.65444054474886627728449087886, 5.20164006899702178333342847264, 5.41676012615338774687665516924, 6.13690005259013972771512187418, 6.21150502900434643631796858377, 6.72598040565036698033673525843, 7.18297506166153508271072468025, 7.74613152589584229848027600152

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