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Properties

Label 4-882e2-1.1-c1e2-0-43
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 3·5-s − 3·6-s + 8-s + 6·9-s − 3·10-s + 3·11-s − 5·13-s + 9·15-s − 16-s + 6·17-s − 6·18-s + 10·19-s − 3·22-s + 3·23-s + 3·24-s + 5·25-s + 5·26-s + 9·27-s + 3·29-s − 9·30-s + 4·31-s + 9·33-s − 6·34-s − 14·37-s − 10·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1.34·5-s − 1.22·6-s + 0.353·8-s + 2·9-s − 0.948·10-s + 0.904·11-s − 1.38·13-s + 2.32·15-s − 1/4·16-s + 1.45·17-s − 1.41·18-s + 2.29·19-s − 0.639·22-s + 0.625·23-s + 0.612·24-s + 25-s + 0.980·26-s + 1.73·27-s + 0.557·29-s − 1.64·30-s + 0.718·31-s + 1.56·33-s − 1.02·34-s − 2.30·37-s − 1.62·38-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: \(0\)
Selberg data: \((4,\ 777924,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.100782575\)
\(L(\frac12)\) \(\approx\) \(4.100782575\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_m
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_au
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.37.o_et
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.43.l_da
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.53.g_el
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_acf
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2^2$ \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.83.d_acw
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.89.abe_pn
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_ads
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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

−9.858940463072466950650768035008, −9.833969949560615810373674139956, −9.490487128299676456524541004090, −9.185203477159087046309975393721, −8.932814381194564121741436678409, −8.230916528597595630748826748176, −7.78640788603460814807806284943, −7.71200222627466327012996492541, −6.94367835896478384319121548641, −6.83249644316601599513517718260, −6.15557701072660678744916743202, −5.27307644448224853393787628935, −5.18138152338355348879699508332, −4.64981015163431147105152991101, −3.66251987396276502763379907594, −3.36968975924260278622053546074, −2.81547609248212821861227976836, −2.30747683987178231865979425522, −1.42417440618217069832351905031, −1.23848234007064724663287252461, 1.23848234007064724663287252461, 1.42417440618217069832351905031, 2.30747683987178231865979425522, 2.81547609248212821861227976836, 3.36968975924260278622053546074, 3.66251987396276502763379907594, 4.64981015163431147105152991101, 5.18138152338355348879699508332, 5.27307644448224853393787628935, 6.15557701072660678744916743202, 6.83249644316601599513517718260, 6.94367835896478384319121548641, 7.71200222627466327012996492541, 7.78640788603460814807806284943, 8.230916528597595630748826748176, 8.932814381194564121741436678409, 9.185203477159087046309975393721, 9.490487128299676456524541004090, 9.833969949560615810373674139956, 9.858940463072466950650768035008

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