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Properties

Label 4-588e2-1.1-c1e2-0-35
Degree $4$
Conductor $345744$
Sign $-1$
Analytic cond. $22.0449$
Root an. cond. $2.16684$
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  + 3-s − 2·9-s + 4·13-s − 2·19-s − 25-s − 5·27-s − 14·31-s − 2·37-s + 4·39-s − 8·43-s − 2·57-s − 2·61-s − 14·67-s − 2·73-s − 75-s − 26·79-s + 81-s − 14·93-s − 20·97-s + 22·103-s − 2·109-s − 2·111-s − 8·117-s − 13·121-s + 127-s − 8·129-s + 131-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s + 1.10·13-s − 0.458·19-s − 1/5·25-s − 0.962·27-s − 2.51·31-s − 0.328·37-s + 0.640·39-s − 1.21·43-s − 0.264·57-s − 0.256·61-s − 1.71·67-s − 0.234·73-s − 0.115·75-s − 2.92·79-s + 1/9·81-s − 1.45·93-s − 2.03·97-s + 2.16·103-s − 0.191·109-s − 0.189·111-s − 0.739·117-s − 1.18·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(345744\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(22.0449\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 345744,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_2$ \( 1 - T + p T^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.a_dt
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.a_bl
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.67.o_hb
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.73.c_fr
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.79.ba_mp
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−8.646784595072977282306137209646, −8.262637018939911096603858847029, −7.51155146227211557820915407148, −7.27657735328498763814943158038, −6.67413798525527807153917855846, −6.02455306491682974275355684196, −5.71711213824960626307579454428, −5.28434328079604079625535316421, −4.44529829557613531132250015524, −3.98660375345473223094112675622, −3.31407617989174490462667281240, −3.04451713874417289677681835886, −2.04448974040096779312937543400, −1.54505867526772913791014367907, 0, 1.54505867526772913791014367907, 2.04448974040096779312937543400, 3.04451713874417289677681835886, 3.31407617989174490462667281240, 3.98660375345473223094112675622, 4.44529829557613531132250015524, 5.28434328079604079625535316421, 5.71711213824960626307579454428, 6.02455306491682974275355684196, 6.67413798525527807153917855846, 7.27657735328498763814943158038, 7.51155146227211557820915407148, 8.262637018939911096603858847029, 8.646784595072977282306137209646

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