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Properties

Label 4-1778112-1.1-c1e2-0-12
Degree $4$
Conductor $1778112$
Sign $-1$
Analytic cond. $113.373$
Root an. cond. $3.26308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 7-s − 3·8-s − 2·11-s − 14-s − 16-s − 2·22-s + 2·23-s + 6·25-s + 28-s + 4·29-s + 5·32-s − 12·37-s + 8·43-s + 2·44-s + 2·46-s + 49-s + 6·50-s − 8·53-s + 3·56-s + 4·58-s + 7·64-s − 4·67-s − 18·71-s − 12·74-s + 2·77-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s − 0.603·11-s − 0.267·14-s − 1/4·16-s − 0.426·22-s + 0.417·23-s + 6/5·25-s + 0.188·28-s + 0.742·29-s + 0.883·32-s − 1.97·37-s + 1.21·43-s + 0.301·44-s + 0.294·46-s + 1/7·49-s + 0.848·50-s − 1.09·53-s + 0.400·56-s + 0.525·58-s + 7/8·64-s − 0.488·67-s − 2.13·71-s − 1.39·74-s + 0.227·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1778112\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(113.373\)
Root analytic conductor: \(3.26308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1778112,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.c_w
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.19.a_abe
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.ac_ac
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ae_bu
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.m_dq
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.41.a_abe
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.47.a_acc
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.i_ec
59$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.59.a_abe
61$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.61.a_ck
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.e_dy
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.s_hq
73$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.73.a_abq
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.m_dq
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.89.a_fq
97$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.97.a_cs
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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

−7.46102221902758854869120286881, −7.16734735530465989728847896423, −6.71685270792997725370193186332, −6.15596140986549181052076649785, −5.85970010218057747254524603782, −5.37766946984049606642364240765, −4.84984327244365570111641729968, −4.64243173452062790099152067411, −4.10797839823755629352669156225, −3.44365101072817236010159430813, −3.04819709247241956913357401999, −2.73611730065490843783404941246, −1.85986834483438245559443500790, −0.955992047237992522914592536688, 0, 0.955992047237992522914592536688, 1.85986834483438245559443500790, 2.73611730065490843783404941246, 3.04819709247241956913357401999, 3.44365101072817236010159430813, 4.10797839823755629352669156225, 4.64243173452062790099152067411, 4.84984327244365570111641729968, 5.37766946984049606642364240765, 5.85970010218057747254524603782, 6.15596140986549181052076649785, 6.71685270792997725370193186332, 7.16734735530465989728847896423, 7.46102221902758854869120286881

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