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Properties

Label 4-882e2-1.1-c1e2-0-25
Degree $4$
Conductor $777924$
Sign $-1$
Analytic cond. $49.6011$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s − 10·11-s + 5·16-s + 20·22-s + 8·23-s − 9·25-s + 10·29-s − 6·32-s − 8·37-s + 4·43-s − 30·44-s − 16·46-s + 18·50-s + 18·53-s − 20·58-s + 7·64-s − 4·67-s − 4·71-s + 16·74-s + 6·79-s − 8·86-s + 40·88-s + 24·92-s − 27·100-s − 36·106-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 3.01·11-s + 5/4·16-s + 4.26·22-s + 1.66·23-s − 9/5·25-s + 1.85·29-s − 1.06·32-s − 1.31·37-s + 0.609·43-s − 4.52·44-s − 2.35·46-s + 2.54·50-s + 2.47·53-s − 2.62·58-s + 7/8·64-s − 0.488·67-s − 0.474·71-s + 1.85·74-s + 0.675·79-s − 0.862·86-s + 4.26·88-s + 2.50·92-s − 2.69·100-s − 3.49·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(49.6011\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 777924,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.11.k_bv
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.29.ak_df
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.43.ae_dm
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.59.a_ad
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.71.e_fq
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.79.ag_gl
83$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.83.a_en
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.a_fp
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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

−8.238671890386923809123178878617, −7.69780043471093435649583790995, −7.21435528320358379927581416164, −7.08055487724091789981125828654, −6.38468865530543948344255312711, −5.71619227739785613107965984191, −5.43475276494163016869526953658, −5.01953856335493271435920798157, −4.36865869014622242250082301422, −3.49120843004488869714011425140, −2.81104469657788282313974403144, −2.57919957316952716799921143645, −1.94506082349214662942482411882, −0.880626761328028180035180249034, 0, 0.880626761328028180035180249034, 1.94506082349214662942482411882, 2.57919957316952716799921143645, 2.81104469657788282313974403144, 3.49120843004488869714011425140, 4.36865869014622242250082301422, 5.01953856335493271435920798157, 5.43475276494163016869526953658, 5.71619227739785613107965984191, 6.38468865530543948344255312711, 7.08055487724091789981125828654, 7.21435528320358379927581416164, 7.69780043471093435649583790995, 8.238671890386923809123178878617

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