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Properties

Label 4-1323e2-1.1-c1e2-0-17
Degree $4$
Conductor $1750329$
Sign $-1$
Analytic cond. $111.602$
Root an. cond. $3.25026$
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  − 2·2-s − 4-s + 8·8-s + 10·11-s − 7·16-s − 20·22-s + 2·23-s − 25-s − 4·29-s − 14·32-s + 6·37-s + 12·43-s − 10·44-s − 4·46-s + 2·50-s − 16·53-s + 8·58-s + 35·64-s − 28·67-s − 14·71-s − 12·74-s − 20·79-s − 24·86-s + 80·88-s − 2·92-s + 100-s + 32·106-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 2.82·8-s + 3.01·11-s − 7/4·16-s − 4.26·22-s + 0.417·23-s − 1/5·25-s − 0.742·29-s − 2.47·32-s + 0.986·37-s + 1.82·43-s − 1.50·44-s − 0.589·46-s + 0.282·50-s − 2.19·53-s + 1.05·58-s + 35/8·64-s − 3.42·67-s − 1.66·71-s − 1.39·74-s − 2.25·79-s − 2.58·86-s + 8.52·88-s − 0.208·92-s + 1/10·100-s + 3.10·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1750329\)    =    \(3^{6} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(111.602\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1750329,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.2.c_f
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.11.ak_bv
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.19.a_bd
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.23.ac_bv
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.a_b
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.53.q_go
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.67.bc_ms
71$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.71.o_hj
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
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$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.a_gn
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.a_by
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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

−7.59315018372019842769294095239, −7.28342071225309798898505652310, −7.13167564560012152294416600100, −6.34215171511087571525667957813, −5.96735414787932054483964871891, −5.63752173484997695290532171122, −4.60087511347345218102319658198, −4.47191624657260288271626711721, −4.22351345808096997888083312699, −3.59632548449757607506915649618, −3.08850093557224289960924465680, −1.92610841189085090066739675799, −1.32164798123200269811823069632, −1.12292208190814267208777122415, 0, 1.12292208190814267208777122415, 1.32164798123200269811823069632, 1.92610841189085090066739675799, 3.08850093557224289960924465680, 3.59632548449757607506915649618, 4.22351345808096997888083312699, 4.47191624657260288271626711721, 4.60087511347345218102319658198, 5.63752173484997695290532171122, 5.96735414787932054483964871891, 6.34215171511087571525667957813, 7.13167564560012152294416600100, 7.28342071225309798898505652310, 7.59315018372019842769294095239

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