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Properties

Label 4-2442e2-1.1-c1e2-0-20
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s − 8·7-s + 3·9-s − 2·11-s − 2·12-s + 16-s − 16·21-s − 6·25-s + 4·27-s + 8·28-s − 4·33-s − 3·36-s − 2·37-s − 20·41-s + 2·44-s + 16·47-s + 2·48-s + 34·49-s − 12·53-s − 24·63-s − 64-s + 8·67-s − 16·71-s − 4·73-s − 12·75-s + 16·77-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 3.02·7-s + 9-s − 0.603·11-s − 0.577·12-s + 1/4·16-s − 3.49·21-s − 6/5·25-s + 0.769·27-s + 1.51·28-s − 0.696·33-s − 1/2·36-s − 0.328·37-s − 3.12·41-s + 0.301·44-s + 2.33·47-s + 0.288·48-s + 34/7·49-s − 1.64·53-s − 3.02·63-s − 1/8·64-s + 0.977·67-s − 1.89·71-s − 0.468·73-s − 1.38·75-s + 1.82·77-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: \(2\)
Selberg data: \((4,\ 5963364,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
37$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.31.a_acg
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.59.a_da
61$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.61.a_cw
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.89.a_abi
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−8.756548328077528180920221813394, −8.624926173230220808284802219517, −7.930320865453923421292795144280, −7.71890607330340851619476618442, −7.11312083535605891530034365982, −6.96340648973417040504454268091, −6.32709397731986096593748348833, −6.29731647240851554488306559609, −5.64317062452078249439641754657, −5.27457426113303192810213900828, −4.71135698874134036670405787441, −3.97566984505735254407087801342, −3.79543703112668604570425583793, −3.44498672825448172100868436560, −2.83890622733191251305237013080, −2.80682950795614099661772242458, −2.05584823633687754317542694068, −1.27829423078473389956973388222, 0, 0, 1.27829423078473389956973388222, 2.05584823633687754317542694068, 2.80682950795614099661772242458, 2.83890622733191251305237013080, 3.44498672825448172100868436560, 3.79543703112668604570425583793, 3.97566984505735254407087801342, 4.71135698874134036670405787441, 5.27457426113303192810213900828, 5.64317062452078249439641754657, 6.29731647240851554488306559609, 6.32709397731986096593748348833, 6.96340648973417040504454268091, 7.11312083535605891530034365982, 7.71890607330340851619476618442, 7.930320865453923421292795144280, 8.624926173230220808284802219517, 8.756548328077528180920221813394

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