Properties

Label 4-676000-1.1-c1e2-0-10
Degree $4$
Conductor $676000$
Sign $1$
Analytic cond. $43.1023$
Root an. cond. $2.56227$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 4·9-s + 10-s + 4·13-s + 16-s + 8·17-s − 4·18-s − 20-s + 25-s − 4·26-s − 32-s − 8·34-s + 4·36-s − 4·37-s + 40-s − 10·41-s − 4·45-s − 2·49-s − 50-s + 4·52-s + 12·61-s + 64-s − 4·65-s + 8·68-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 4/3·9-s + 0.316·10-s + 1.10·13-s + 1/4·16-s + 1.94·17-s − 0.942·18-s − 0.223·20-s + 1/5·25-s − 0.784·26-s − 0.176·32-s − 1.37·34-s + 2/3·36-s − 0.657·37-s + 0.158·40-s − 1.56·41-s − 0.596·45-s − 2/7·49-s − 0.141·50-s + 0.554·52-s + 1.53·61-s + 1/8·64-s − 0.496·65-s + 0.970·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676000\)    =    \(2^{5} \cdot 5^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(43.1023\)
Root analytic conductor: \(2.56227\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.695226293\)
\(L(\frac12)\) \(\approx\) \(1.695226293\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$ \( 1 + T \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ai_bu
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.23.a_ai
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.31.a_w
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.k_du
43$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.43.a_abo
47$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.47.a_cw
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.59.a_bi
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.am_fm
67$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.67.a_ade
71$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.71.a_adu
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.i_gc
79$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.79.a_ack
83$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.83.a_aeg
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.ae_bm
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.ag_es
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.457181634898399983693068225617, −7.916666675576647089006722103456, −7.43853746801600000652128296728, −7.12581799617784737309152339580, −6.76336662103436792545248132150, −6.06281987065218553662680035524, −5.78738231467373749304149939136, −5.06239667851920461091068473214, −4.66302546803605621403004769055, −3.87597664984406860773059991579, −3.49964744278714463856022753990, −3.13410972088078058967844565146, −2.03821321210769053371398765842, −1.44663761485832477617108935568, −0.812783200067965994526159777954, 0.812783200067965994526159777954, 1.44663761485832477617108935568, 2.03821321210769053371398765842, 3.13410972088078058967844565146, 3.49964744278714463856022753990, 3.87597664984406860773059991579, 4.66302546803605621403004769055, 5.06239667851920461091068473214, 5.78738231467373749304149939136, 6.06281987065218553662680035524, 6.76336662103436792545248132150, 7.12581799617784737309152339580, 7.43853746801600000652128296728, 7.916666675576647089006722103456, 8.457181634898399983693068225617

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