Properties

Label 4-1152e2-1.1-c1e2-0-33
Degree $4$
Conductor $1327104$
Sign $1$
Analytic cond. $84.6173$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·17-s + 6·25-s + 16·41-s − 14·49-s − 12·73-s + 32·89-s + 36·97-s − 32·113-s + 22·121-s + ⋯
L(s)  = 1  + 3.88·17-s + 6/5·25-s + 2.49·41-s − 2·49-s − 1.40·73-s + 3.39·89-s + 3.65·97-s − 3.01·113-s + 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1327104\)    =    \(2^{14} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(84.6173\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1327104,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.806782856\)
\(L(\frac12)\) \(\approx\) \(2.806782856\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.17.aq_du
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.41.aq_fq
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.67.a_afe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.89.abg_qs
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.97.abk_ty
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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

−10.13852419384693859791938650779, −9.569195906209353316386103391816, −9.154023296156999509157275982202, −8.992160693593646143980419697927, −8.054031608146775213709927224735, −8.032695675415850048715968186992, −7.52058281928250969497260568069, −7.38636052496249254937044044400, −6.57431551751645437142274452492, −6.20441728924070196660451788414, −5.70426406218297600320158904448, −5.43560605668167663455910920234, −4.85520788150980356817474621153, −4.48912698965887130754000845781, −3.62316112592333676467084540495, −3.33587521588229330439508665186, −2.96975377663197412870246570883, −2.17637646862648988184829450490, −1.23654910196504229459775032339, −0.883939896211164024581406165787, 0.883939896211164024581406165787, 1.23654910196504229459775032339, 2.17637646862648988184829450490, 2.96975377663197412870246570883, 3.33587521588229330439508665186, 3.62316112592333676467084540495, 4.48912698965887130754000845781, 4.85520788150980356817474621153, 5.43560605668167663455910920234, 5.70426406218297600320158904448, 6.20441728924070196660451788414, 6.57431551751645437142274452492, 7.38636052496249254937044044400, 7.52058281928250969497260568069, 8.032695675415850048715968186992, 8.054031608146775213709927224735, 8.992160693593646143980419697927, 9.154023296156999509157275982202, 9.569195906209353316386103391816, 10.13852419384693859791938650779

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