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Properties

Label 4-238e2-1.1-c1e2-0-8
Degree $4$
Conductor $56644$
Sign $1$
Analytic cond. $3.61167$
Root an. cond. $1.37856$
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  + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 2·9-s + 12·11-s − 8·14-s + 5·16-s − 4·18-s + 24·22-s − 10·25-s − 12·28-s + 6·32-s − 6·36-s − 8·37-s + 16·43-s + 36·44-s + 9·49-s − 20·50-s − 12·53-s − 16·56-s + 8·63-s + 7·64-s + 16·67-s − 8·72-s − 16·74-s − 48·77-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 2/3·9-s + 3.61·11-s − 2.13·14-s + 5/4·16-s − 0.942·18-s + 5.11·22-s − 2·25-s − 2.26·28-s + 1.06·32-s − 36-s − 1.31·37-s + 2.43·43-s + 5.42·44-s + 9/7·49-s − 2.82·50-s − 1.64·53-s − 2.13·56-s + 1.00·63-s + 7/8·64-s + 1.95·67-s − 0.942·72-s − 1.85·74-s − 5.47·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56644\)    =    \(2^{2} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(3.61167\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.167608159\)
\(L(\frac12)\) \(\approx\) \(3.167608159\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−9.795849732674491714602252272054, −9.435449454754526833910951528272, −9.286071988264369821451329617999, −8.550482821819457584609282080376, −7.75961744967946030738916282437, −6.98368217520104403156128027372, −6.65330731261113455462227374582, −6.16055468076817207109148001002, −5.99911855171913780844071238816, −5.16488039277883802975012518677, −4.04100934533934329070442787144, −3.90229547122613769308947697962, −3.50747685387131529534531471772, −2.52652883309992049334898398402, −1.44781063014612234378860428910, 1.44781063014612234378860428910, 2.52652883309992049334898398402, 3.50747685387131529534531471772, 3.90229547122613769308947697962, 4.04100934533934329070442787144, 5.16488039277883802975012518677, 5.99911855171913780844071238816, 6.16055468076817207109148001002, 6.65330731261113455462227374582, 6.98368217520104403156128027372, 7.75961744967946030738916282437, 8.550482821819457584609282080376, 9.286071988264369821451329617999, 9.435449454754526833910951528272, 9.795849732674491714602252272054

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