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Properties

Label 4-882e2-1.1-c1e2-0-51
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 + 2·5-s + 16-s − 6·17-s + 2·20-s − 3·25-s − 10·37-s − 16·41-s − 10·43-s + 14·47-s + 16·59-s + 64-s + 4·67-s − 6·68-s − 12·79-s + 2·80-s − 12·83-s − 12·85-s − 16·89-s − 3·100-s + 4·101-s + 4·109-s − 13·121-s − 10·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.894·5-s + 1/4·16-s − 1.45·17-s + 0.447·20-s − 3/5·25-s − 1.64·37-s − 2.49·41-s − 1.52·43-s + 2.04·47-s + 2.08·59-s + 1/8·64-s + 0.488·67-s − 0.727·68-s − 1.35·79-s + 0.223·80-s − 1.31·83-s − 1.30·85-s − 1.69·89-s − 0.299·100-s + 0.398·101-s + 0.383·109-s − 1.18·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
7 \( 1 \)
good5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.ac_h
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.g_bq
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.29.a_av
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.a_at
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.k_du
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.q_fm
43$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.k_dy
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.47.ao_eo
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.53.a_az
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.59.aq_gz
61$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.61.a_acs
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.ae_bm
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.73.a_da
79$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.79.m_gn
83$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.83.m_dd
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.89.q_ji
97$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \) 2.97.a_abx
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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

−8.245434911163522179942131896458, −7.40654818788603568123567819672, −6.99252573337364764107289304698, −6.78612548464342363545429212646, −6.35001468641052098851909960781, −5.64272634046843917739300992251, −5.48239853492490009131896414899, −4.92062288672442932083282428879, −4.27355941636326996832260805751, −3.74253949199601787379469790972, −3.16200492318022523532089864977, −2.41038112635067118788471646762, −1.99609970980440244853994942412, −1.43938001501524395652906031811, 0, 1.43938001501524395652906031811, 1.99609970980440244853994942412, 2.41038112635067118788471646762, 3.16200492318022523532089864977, 3.74253949199601787379469790972, 4.27355941636326996832260805751, 4.92062288672442932083282428879, 5.48239853492490009131896414899, 5.64272634046843917739300992251, 6.35001468641052098851909960781, 6.78612548464342363545429212646, 6.99252573337364764107289304698, 7.40654818788603568123567819672, 8.245434911163522179942131896458

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