↑

Properties

Label 4-1078e2-1.1-c1e2-0-6
Degree $4$
Conductor $1162084$
Sign $1$
Analytic cond. $74.0954$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s − 6·9-s + 2·11-s + 5·16-s + 12·18-s − 4·22-s + 12·23-s − 2·25-s + 16·29-s − 6·32-s − 18·36-s − 12·37-s + 20·43-s + 6·44-s − 24·46-s + 4·50-s + 12·53-s − 32·58-s + 7·64-s − 8·67-s + 24·72-s + 24·74-s + 27·81-s − 40·86-s − 8·88-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2·9-s + 0.603·11-s + 5/4·16-s + 2.82·18-s − 0.852·22-s + 2.50·23-s − 2/5·25-s + 2.97·29-s − 1.06·32-s − 3·36-s − 1.97·37-s + 3.04·43-s + 0.904·44-s − 3.53·46-s + 0.565·50-s + 1.64·53-s − 4.20·58-s + 7/8·64-s − 0.977·67-s + 2.82·72-s + 2.78·74-s + 3·81-s − 4.31·86-s − 0.852·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1162084\)    =    \(2^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(74.0954\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1162084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9984116214\)
\(L(\frac12)\) \(\approx\) \(0.9984116214\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.17.a_ba
19$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \) 2.19.a_bk
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.29.aq_es
31$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \) 2.31.a_ci
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.41.a_k
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.47.a_bs
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \) 2.61.a_ea
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.73.a_cw
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \) 2.83.a_em
89$C_2^2$ \( 1 - 160 T^{2} + p^{2} T^{4} \) 2.89.a_age
97$C_2^2$ \( 1 + 192 T^{2} + p^{2} T^{4} \) 2.97.a_hk
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.898290459616744599754825348698, −9.731916096058635914496271000219, −8.927719624716285309711938621442, −8.774048791834705073474445509367, −8.659357197050104971143000220567, −8.411640667493256185691590438354, −7.42266056369151013578842586709, −7.42041539660764684964341793639, −6.90598034759782760499441213991, −6.30898908876404189734012114747, −5.98642626268014074580496908564, −5.64558750702522696840066099678, −4.86406830789111300840395276983, −4.64171572849634868423435587742, −3.44685398696926283075448117350, −3.29742834615015440839675956396, −2.54473546694734854155889576987, −2.32109854929071795135534532210, −1.09973017025046616832861966335, −0.67651464646677469676777367322, 0.67651464646677469676777367322, 1.09973017025046616832861966335, 2.32109854929071795135534532210, 2.54473546694734854155889576987, 3.29742834615015440839675956396, 3.44685398696926283075448117350, 4.64171572849634868423435587742, 4.86406830789111300840395276983, 5.64558750702522696840066099678, 5.98642626268014074580496908564, 6.30898908876404189734012114747, 6.90598034759782760499441213991, 7.42041539660764684964341793639, 7.42266056369151013578842586709, 8.411640667493256185691590438354, 8.659357197050104971143000220567, 8.774048791834705073474445509367, 8.927719624716285309711938621442, 9.731916096058635914496271000219, 9.898290459616744599754825348698

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