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Properties

Label 4-119936-1.1-c1e2-0-4
Degree $4$
Conductor $119936$
Sign $-1$
Analytic cond. $7.64722$
Root an. cond. $1.66293$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 5·9-s + 14-s + 16-s − 16·17-s − 5·18-s + 3·23-s − 6·25-s + 28-s + 32-s − 16·34-s − 5·36-s − 41-s + 3·46-s + 15·47-s − 13·49-s − 6·50-s + 56-s − 5·63-s + 64-s − 16·68-s + 3·71-s − 5·72-s − 5·79-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 5/3·9-s + 0.267·14-s + 1/4·16-s − 3.88·17-s − 1.17·18-s + 0.625·23-s − 6/5·25-s + 0.188·28-s + 0.176·32-s − 2.74·34-s − 5/6·36-s − 0.156·41-s + 0.442·46-s + 2.18·47-s − 1.85·49-s − 0.848·50-s + 0.133·56-s − 0.629·63-s + 1/8·64-s − 1.94·68-s + 0.356·71-s − 0.589·72-s − 0.562·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(119936\)    =    \(2^{7} \cdot 937\)
Sign: $-1$
Analytic conductor: \(7.64722\)
Root analytic conductor: \(1.66293\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 119936,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
937$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 23 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ab_o
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.13.a_as
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.17.q_du
19$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.19.a_b
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.ad_ai
29$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.29.a_z
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.37.a_abd
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.41.b_ck
43$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.43.a_abi
47$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.47.ap_fi
53$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.53.a_br
59$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \) 2.59.a_ci
61$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.61.a_acv
67$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.67.a_s
71$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.ad_fc
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.a_dt
79$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.79.f_fw
83$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.83.a_ak
89$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.89.v_lc
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.97.al_hw
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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

−9.079785362007829535833068165222, −8.741806748240636291302783285136, −8.280352340020058681746602368520, −7.78528556864084178439589457919, −6.94530762324739907560732818060, −6.68739751872016380017770073196, −6.12376735543076667140697226277, −5.61317951408748853990831553036, −5.04763977091164995874420812171, −4.38421543921979723002939349143, −4.08361128140497311043616053116, −3.08204225192201180511105926760, −2.45709152366038507689664067368, −1.96767531937932045488436170771, 0, 1.96767531937932045488436170771, 2.45709152366038507689664067368, 3.08204225192201180511105926760, 4.08361128140497311043616053116, 4.38421543921979723002939349143, 5.04763977091164995874420812171, 5.61317951408748853990831553036, 6.12376735543076667140697226277, 6.68739751872016380017770073196, 6.94530762324739907560732818060, 7.78528556864084178439589457919, 8.280352340020058681746602368520, 8.741806748240636291302783285136, 9.079785362007829535833068165222

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