↑

Properties

Label 4-244000-1.1-c1e2-0-8
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 $1$

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 + 2·17-s − 4·18-s − 20-s + 25-s − 4·26-s + 10·29-s + 32-s + 2·34-s − 4·36-s − 2·37-s − 40-s − 20·41-s + 4·45-s + 2·49-s + 50-s − 4·52-s − 4·53-s + 10·58-s − 3·61-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 + 0.485·17-s − 0.942·18-s − 0.223·20-s + 1/5·25-s − 0.784·26-s + 1.85·29-s + 0.176·32-s + 0.342·34-s − 2/3·36-s − 0.328·37-s − 0.158·40-s − 3.12·41-s + 0.596·45-s + 2/7·49-s + 0.141·50-s − 0.554·52-s − 0.549·53-s + 1.31·58-s − 0.384·61-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: \(1\)
Selberg data: \((4,\ 244000,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
5$C_1$ \( 1 + T \)
61$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ac_bi
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.23.a_ao
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.ak_cw
31$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.31.a_acc
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.c_ag
41$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.u_gw
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.47.a_bg
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.e_bu
59$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.59.a_ac
67$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.67.a_abu
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.79.a_ade
83$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.83.a_cm
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.m_hq
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.m_ig
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

−8.527565529095868816048342706857, −8.187229329596449208044209671130, −7.950787979002723920598482480648, −7.10325659015740832284385881109, −6.81857814444748456016128917287, −6.36497780905053588819064830733, −5.64574001916204253028043540297, −5.20759512886621870684999313441, −4.88622676193268438673788388900, −4.21376004236382692181015334550, −3.51050579405019984920683309967, −2.91943247778741616189395744665, −2.59479684280361760187893557549, −1.49959183140170230867147282267, 0, 1.49959183140170230867147282267, 2.59479684280361760187893557549, 2.91943247778741616189395744665, 3.51050579405019984920683309967, 4.21376004236382692181015334550, 4.88622676193268438673788388900, 5.20759512886621870684999313441, 5.64574001916204253028043540297, 6.36497780905053588819064830733, 6.81857814444748456016128917287, 7.10325659015740832284385881109, 7.950787979002723920598482480648, 8.187229329596449208044209671130, 8.527565529095868816048342706857

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