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Properties

Label 4-555579-1.1-c1e2-0-5
Degree $4$
Conductor $555579$
Sign $1$
Analytic cond. $35.4241$
Root an. cond. $2.43963$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·7-s − 4·16-s + 19-s + 25-s − 2·43-s + 61·49-s + 2·61-s − 22·73-s − 40·112-s + 21·121-s + 127-s + 131-s + 10·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 10·175-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 3.77·7-s − 16-s + 0.229·19-s + 1/5·25-s − 0.304·43-s + 61/7·49-s + 0.256·61-s − 2.57·73-s − 3.77·112-s + 1.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.867·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.755·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(555579\)    =    \(3^{4} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(35.4241\)
Root analytic conductor: \(2.43963\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 555579,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.259932801\)
\(L(\frac12)\) \(\approx\) \(3.259932801\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
19$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.7.ak_bn
11$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.11.a_av
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.17.a_abh
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.47.a_an
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.61.ac_et
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.a_dq
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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

−8.457707288259077960197054760580, −8.073957841478482575776375446457, −7.63963131698664278720921224669, −7.17716072499260981468870948681, −6.93848179848514410598268972312, −5.87826513264342669434687011041, −5.65732719817381090492939531705, −4.98770944705006228649621282628, −4.66107001383131332856158508778, −4.47368714118890317110419998334, −3.83177545151462000540357703330, −2.81842226367277386936517101633, −2.07794009677933110260559141292, −1.73227044111605436033558510736, −1.04281368568489569781280600938, 1.04281368568489569781280600938, 1.73227044111605436033558510736, 2.07794009677933110260559141292, 2.81842226367277386936517101633, 3.83177545151462000540357703330, 4.47368714118890317110419998334, 4.66107001383131332856158508778, 4.98770944705006228649621282628, 5.65732719817381090492939531705, 5.87826513264342669434687011041, 6.93848179848514410598268972312, 7.17716072499260981468870948681, 7.63963131698664278720921224669, 8.073957841478482575776375446457, 8.457707288259077960197054760580

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