Properties

Label 4-676000-1.1-c1e2-0-25
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s − 9-s + 10-s + 2·13-s + 16-s + 3·17-s + 18-s − 20-s + 25-s − 2·26-s + 3·29-s − 32-s − 3·34-s − 36-s − 7·37-s + 40-s + 3·41-s + 45-s + 5·49-s − 50-s + 2·52-s − 12·53-s − 3·58-s − 28·61-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 + 0.554·13-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.223·20-s + 1/5·25-s − 0.392·26-s + 0.557·29-s − 0.176·32-s − 0.514·34-s − 1/6·36-s − 1.15·37-s + 0.158·40-s + 0.468·41-s + 0.149·45-s + 5/7·49-s − 0.141·50-s + 0.277·52-s − 1.64·53-s − 0.393·58-s − 3.58·61-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: \(1\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
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$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.11.a_ab
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ad_bi
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.19.a_bd
23$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.23.a_aq
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ad_cg
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.37.h_dg
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.ad_cm
43$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.43.a_r
47$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \) 2.47.a_bx
53$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.m_fd
59$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.59.a_r
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.61.bc_mg
67$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.67.a_ao
71$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \) 2.71.a_em
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.ac_fi
79$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.79.a_acn
83$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.83.a_bo
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.s_jq
97$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.ae_en
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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

−8.105630465279676568065200103399, −7.75500375629573116937004093273, −7.32405764960486285969670034765, −6.91672182871210589798164049597, −6.30679328627392989362577619248, −5.91891950258160904250663824116, −5.55069658304606360009438427803, −4.72946633360835778285371042857, −4.46722529202266215215785044729, −3.58741430792992342794029026612, −3.23021728427081232010672472120, −2.69942176364012743511866030439, −1.77528514736944221016048597946, −1.14335518199776758983685468582, 0, 1.14335518199776758983685468582, 1.77528514736944221016048597946, 2.69942176364012743511866030439, 3.23021728427081232010672472120, 3.58741430792992342794029026612, 4.46722529202266215215785044729, 4.72946633360835778285371042857, 5.55069658304606360009438427803, 5.91891950258160904250663824116, 6.30679328627392989362577619248, 6.91672182871210589798164049597, 7.32405764960486285969670034765, 7.75500375629573116937004093273, 8.105630465279676568065200103399

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