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Properties

Label 4-2442e2-1.1-c1e2-0-1
Degree $4$
Conductor $5963364$
Sign $1$
Analytic cond. $380.229$
Root an. cond. $4.41582$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4-s − 8·5-s + 3·9-s + 5·11-s − 2·12-s + 16·15-s + 16-s − 8·20-s + 18·23-s + 38·25-s − 4·27-s − 4·31-s − 10·33-s + 3·36-s + 2·37-s + 5·44-s − 24·45-s − 20·47-s − 2·48-s − 5·49-s + 6·53-s − 40·55-s − 8·59-s + 16·60-s + 64-s + 12·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 3.57·5-s + 9-s + 1.50·11-s − 0.577·12-s + 4.13·15-s + 1/4·16-s − 1.78·20-s + 3.75·23-s + 38/5·25-s − 0.769·27-s − 0.718·31-s − 1.74·33-s + 1/2·36-s + 0.328·37-s + 0.753·44-s − 3.57·45-s − 2.91·47-s − 0.288·48-s − 5/7·49-s + 0.824·53-s − 5.39·55-s − 1.04·59-s + 2.06·60-s + 1/8·64-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5963364\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(380.229\)
Root analytic conductor: \(4.41582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5963364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5453686450\)
\(L(\frac12)\) \(\approx\) \(0.5453686450\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( ( 1 + T )^{2} \)
11$C_2$ \( 1 - 5 T + p T^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.5.i_ba
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.a_r
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.23.as_ex
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.47.u_hm
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.67.am_go
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.a_ax
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.79.a_es
83$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.83.a_fl
89$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.89.aw_ln
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.97.am_iw
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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

−7.32323094324505148733163418443, −6.89342426467112599170590939233, −6.50722784968318237501983344506, −6.42380409964038574403615594445, −5.49573142340352003561742069236, −4.91417472073856393922961060427, −4.69516264772634405387664054544, −4.54153848033705011488514608460, −3.69413280585934281517510849684, −3.62249531038962811542206939005, −3.25642768509079454232693835647, −2.71542071497926879083795377763, −1.31585175497619613956901167226, −1.13658449571290171220917545956, −0.34224393506567788777553888401, 0.34224393506567788777553888401, 1.13658449571290171220917545956, 1.31585175497619613956901167226, 2.71542071497926879083795377763, 3.25642768509079454232693835647, 3.62249531038962811542206939005, 3.69413280585934281517510849684, 4.54153848033705011488514608460, 4.69516264772634405387664054544, 4.91417472073856393922961060427, 5.49573142340352003561742069236, 6.42380409964038574403615594445, 6.50722784968318237501983344506, 6.89342426467112599170590939233, 7.32323094324505148733163418443

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