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Properties

Label 4-244000-1.1-c1e2-0-3
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 − 9-s − 10-s + 8·13-s + 16-s + 3·17-s − 18-s − 20-s + 25-s + 8·26-s − 12·29-s + 32-s + 3·34-s − 36-s + 15·37-s − 40-s + 8·41-s + 45-s + 5·49-s + 50-s + 8·52-s − 7·53-s − 12·58-s − 6·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 1/3·9-s − 0.316·10-s + 2.21·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.223·20-s + 1/5·25-s + 1.56·26-s − 2.22·29-s + 0.176·32-s + 0.514·34-s − 1/6·36-s + 2.46·37-s − 0.158·40-s + 1.24·41-s + 0.149·45-s + 5/7·49-s + 0.141·50-s + 1.10·52-s − 0.961·53-s − 1.57·58-s − 0.768·61-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: \(0\)
Selberg data: \((4,\ 244000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.809243548\)
\(L(\frac12)\) \(\approx\) \(2.809243548\)
\(L(\frac{3}{2})\) 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$ \( ( 1 + T )( 1 + 5 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.13.ai_bh
17$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ad_y
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.23.a_abm
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.m_da
31$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.31.a_acc
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.37.ap_eu
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ai_de
43$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.43.a_abo
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.47.a_ac
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.h_dk
59$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \) 2.59.a_cz
67$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.67.a_bc
71$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.71.a_dy
73$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.n_fc
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.83.a_adt
89$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ah_gw
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ak_go
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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

−8.994497812985441097333186800527, −8.410993189842138354379493724423, −7.920107335637643954652603019738, −7.56722821389737554323162603261, −7.14491421732180858204746519104, −6.25664325389821886307296895811, −5.98273371528139647887072331570, −5.77905580425571720051962727268, −5.03296346782038614857880832288, −4.17424845101585418813451282030, −4.03942128801727988319817639778, −3.32830085304161878826311635965, −2.86729956486879638370060810552, −1.84842562542878735930328727715, −0.989768776983117757485266874984, 0.989768776983117757485266874984, 1.84842562542878735930328727715, 2.86729956486879638370060810552, 3.32830085304161878826311635965, 4.03942128801727988319817639778, 4.17424845101585418813451282030, 5.03296346782038614857880832288, 5.77905580425571720051962727268, 5.98273371528139647887072331570, 6.25664325389821886307296895811, 7.14491421732180858204746519104, 7.56722821389737554323162603261, 7.920107335637643954652603019738, 8.410993189842138354379493724423, 8.994497812985441097333186800527

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