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Properties

Label 4-882e2-1.1-c1e2-0-36
Degree $4$
Conductor $777924$
Sign $-1$
Analytic cond. $49.6011$
Root an. cond. $2.65382$
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  − 4-s + 16-s − 25-s − 16·37-s + 20·43-s − 64-s − 20·67-s − 2·79-s + 100-s − 28·109-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s − 20·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/4·16-s − 1/5·25-s − 2.63·37-s + 3.04·43-s − 1/8·64-s − 2.44·67-s − 0.225·79-s + 1/10·100-s − 2.68·109-s + 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s − 1.52·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-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: yes
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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
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_abu
29$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.29.a_x
31$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.31.a_cj
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.53.a_adt
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.a_ef
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.67.u_ja
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.71.a_aec
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.a_dh
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.a_hl
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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

−8.082439908276773874714430944766, −7.45875487611721098599084079839, −7.33913456974674757946977572605, −6.73543926276859007978207337344, −6.17592201997476883493564675084, −5.71218349744810749148047423737, −5.36558710631371762946043955779, −4.77144739920261025559248514602, −4.27801691901451199753591670015, −3.84551802497974398067311410206, −3.23930263678038188203067993870, −2.65600747611692068912230262628, −1.90983840937733297645803308865, −1.13677116756828550012468117580, 0, 1.13677116756828550012468117580, 1.90983840937733297645803308865, 2.65600747611692068912230262628, 3.23930263678038188203067993870, 3.84551802497974398067311410206, 4.27801691901451199753591670015, 4.77144739920261025559248514602, 5.36558710631371762946043955779, 5.71218349744810749148047423737, 6.17592201997476883493564675084, 6.73543926276859007978207337344, 7.33913456974674757946977572605, 7.45875487611721098599084079839, 8.082439908276773874714430944766

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