Properties

Label 4-968e2-1.1-c1e2-0-10
Degree $4$
Conductor $937024$
Sign $-1$
Analytic cond. $59.7454$
Root an. cond. $2.78020$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4·3-s − 4-s + 4·6-s − 4·7-s − 3·8-s + 6·9-s − 4·12-s + 2·13-s − 4·14-s − 16-s + 6·18-s − 16·21-s − 12·24-s − 9·25-s + 2·26-s − 4·27-s + 4·28-s + 18·29-s + 5·32-s − 6·36-s + 8·39-s − 16·42-s − 4·48-s − 2·49-s − 9·50-s − 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.30·3-s − 1/2·4-s + 1.63·6-s − 1.51·7-s − 1.06·8-s + 2·9-s − 1.15·12-s + 0.554·13-s − 1.06·14-s − 1/4·16-s + 1.41·18-s − 3.49·21-s − 2.44·24-s − 9/5·25-s + 0.392·26-s − 0.769·27-s + 0.755·28-s + 3.34·29-s + 0.883·32-s − 36-s + 1.28·39-s − 2.46·42-s − 0.577·48-s − 2/7·49-s − 1.27·50-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(937024\)    =    \(2^{6} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(59.7454\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 937024,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
11 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.17.a_j
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.29.as_fj
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.a_cn
41$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.41.a_cf
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.59.aq_ha
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.97.ba_nz
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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

−8.191754991883427916137459483684, −7.81304245413044943247521419550, −6.86022704125109365573887961245, −6.72221402891887535126565651102, −6.23631640550960698833745661801, −5.51160152763630926450656211881, −5.38415572352897256696127903868, −4.31690254498802738209818156699, −3.87299050839595889181889379856, −3.84177902362047353880486406829, −3.01368167787977742950054040757, −2.75274084463573300677827364197, −2.55123860245483446996298704576, −1.38763061579404254635807455108, 0, 1.38763061579404254635807455108, 2.55123860245483446996298704576, 2.75274084463573300677827364197, 3.01368167787977742950054040757, 3.84177902362047353880486406829, 3.87299050839595889181889379856, 4.31690254498802738209818156699, 5.38415572352897256696127903868, 5.51160152763630926450656211881, 6.23631640550960698833745661801, 6.72221402891887535126565651102, 6.86022704125109365573887961245, 7.81304245413044943247521419550, 8.191754991883427916137459483684

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