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Properties

Label 4-244000-1.1-c1e2-0-1
Degree $4$
Conductor $244000$
Sign $1$
Analytic cond. $15.5576$
Root an. cond. $1.98603$
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  + 2-s + 4-s + 5-s + 8-s − 5·9-s + 10-s − 2·13-s + 16-s + 6·17-s − 5·18-s + 20-s + 25-s − 2·26-s + 32-s + 6·34-s − 5·36-s + 6·37-s + 40-s + 4·41-s − 5·45-s + 5·49-s + 50-s − 2·52-s + 3·53-s + 8·61-s + 64-s − 2·65-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 5/3·9-s + 0.316·10-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 1.17·18-s + 0.223·20-s + 1/5·25-s − 0.392·26-s + 0.176·32-s + 1.02·34-s − 5/6·36-s + 0.986·37-s + 0.158·40-s + 0.624·41-s − 0.745·45-s + 5/7·49-s + 0.141·50-s − 0.277·52-s + 0.412·53-s + 1.02·61-s + 1/8·64-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 244000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(244000\)    =    \(2^{5} \cdot 5^{3} \cdot 61\)
Sign: $1$
Analytic conductor: \(15.5576\)
Root analytic conductor: \(1.98603\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 244000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.616191120\)
\(L(\frac12)\) \(\approx\) \(2.616191120\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
61$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 7 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.c_c
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ag_s
19$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.19.a_abg
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.a_abj
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.31.a_i
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ag_cg
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.ae_ao
43$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.43.a_be
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.ad_ca
59$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.59.a_aw
67$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.67.a_dm
71$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \) 2.71.a_acu
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.73.ad_fm
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.83.a_cs
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.89.ap_iu
97$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.ab_bm
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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

−9.011227075849805237199541055571, −8.397799064425679481599772812178, −8.038135196859620144919153239691, −7.47469561842956477869441144182, −7.11123362584789059939205655808, −6.26974027218379243260555284436, −5.99949426210992173420802220266, −5.58220327698708146300896285770, −5.11219183220872294708158941364, −4.61863158153176632124363465661, −3.73850218184877365387545489736, −3.28628895350517543628589564353, −2.62169861211384967508479258406, −2.17617099368752893709168079321, −0.891444241440217796957538202627, 0.891444241440217796957538202627, 2.17617099368752893709168079321, 2.62169861211384967508479258406, 3.28628895350517543628589564353, 3.73850218184877365387545489736, 4.61863158153176632124363465661, 5.11219183220872294708158941364, 5.58220327698708146300896285770, 5.99949426210992173420802220266, 6.26974027218379243260555284436, 7.11123362584789059939205655808, 7.47469561842956477869441144182, 8.038135196859620144919153239691, 8.397799064425679481599772812178, 9.011227075849805237199541055571

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