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Properties

Label 4-1220000-1.1-c1e2-0-3
Degree $4$
Conductor $1220000$
Sign $1$
Analytic cond. $77.7882$
Root an. cond. $2.96980$
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 − 8-s − 9-s + 4·13-s + 16-s + 13·17-s + 18-s − 4·26-s − 8·29-s − 32-s − 13·34-s − 36-s − 7·37-s + 4·41-s + 11·49-s + 4·52-s + 13·53-s + 8·58-s + 6·61-s + 64-s + 13·68-s + 72-s − 15·73-s + 7·74-s − 8·81-s − 4·82-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1/3·9-s + 1.10·13-s + 1/4·16-s + 3.15·17-s + 0.235·18-s − 0.784·26-s − 1.48·29-s − 0.176·32-s − 2.22·34-s − 1/6·36-s − 1.15·37-s + 0.624·41-s + 11/7·49-s + 0.554·52-s + 1.78·53-s + 1.05·58-s + 0.768·61-s + 1/8·64-s + 1.57·68-s + 0.117·72-s − 1.75·73-s + 0.813·74-s − 8/9·81-s − 0.441·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1220000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1220000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1220000\)    =    \(2^{5} \cdot 5^{4} \cdot 61\)
Sign: $1$
Analytic conductor: \(77.7882\)
Root analytic conductor: \(2.96980\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1220000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771485123\)
\(L(\frac12)\) \(\approx\) \(1.771485123\)
\(L(\frac{3}{2})\) 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 \( 1 \)
61$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 7 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.ae_v
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.17.an_cy
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
31$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.31.a_y
37$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.h_co
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ae_de
43$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.43.a_bu
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.47.a_acs
53$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + p T^{2} ) \) 2.53.an_ec
59$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.59.a_cd
67$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.67.a_dc
71$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.71.a_aco
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.p_gq
79$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.79.a_abi
83$C_2^2$ \( 1 + 139 T^{2} + p^{2} T^{4} \) 2.83.a_fj
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.89.j_ga
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.97.ay_mw
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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

−7.944079674215710174083705133495, −7.63761304802003832925943840675, −7.24058783417139843852626336010, −6.98190521838698117581077135159, −5.98560491539327152323768292395, −5.94033187236765858848701191612, −5.53317158747977774958162474746, −5.13911069257074279782913647420, −4.24058978456206252937907113215, −3.58749928329521382200856757956, −3.48179184562867865627845708559, −2.80827500533861491181573715803, −2.02906101519760742195143451319, −1.31909499726767759200852004362, −0.74753170431895483070031347213, 0.74753170431895483070031347213, 1.31909499726767759200852004362, 2.02906101519760742195143451319, 2.80827500533861491181573715803, 3.48179184562867865627845708559, 3.58749928329521382200856757956, 4.24058978456206252937907113215, 5.13911069257074279782913647420, 5.53317158747977774958162474746, 5.94033187236765858848701191612, 5.98560491539327152323768292395, 6.98190521838698117581077135159, 7.24058783417139843852626336010, 7.63761304802003832925943840675, 7.944079674215710174083705133495

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