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Properties

Label 4-676000-1.1-c1e2-0-27
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 − 4·13-s + 16-s − 5·17-s − 18-s + 20-s + 25-s + 4·26-s + 3·29-s − 32-s + 5·34-s + 36-s + 5·37-s − 40-s + 3·41-s + 45-s + 5·49-s − 50-s − 4·52-s − 18·53-s − 3·58-s + 10·61-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 − 1.10·13-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.223·20-s + 1/5·25-s + 0.784·26-s + 0.557·29-s − 0.176·32-s + 0.857·34-s + 1/6·36-s + 0.821·37-s − 0.158·40-s + 0.468·41-s + 0.149·45-s + 5/7·49-s − 0.141·50-s − 0.554·52-s − 2.47·53-s − 0.393·58-s + 1.28·61-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$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.f_bo
19$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.19.a_bj
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.29.ad_cc
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.af_by
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.41.ad_m
43$C_2^2$ \( 1 - 69 T^{2} + p^{2} T^{4} \) 2.43.a_acr
47$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.47.a_f
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.59.a_az
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.ak_es
67$C_2^2$ \( 1 - 100 T^{2} + p^{2} T^{4} \) 2.67.a_adw
71$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.71.a_bo
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.m_de
79$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \) 2.79.a_cp
83$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.83.a_abo
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ag_gw
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.97.au_jl
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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

−8.107051730609216110833704749959, −7.67181221204779541731261353120, −7.30258037467520962079345595243, −6.80985712973342974042195712008, −6.36438060854535008496393142433, −6.07765952753275119236893797977, −5.30581700616829522450723512615, −4.91465544529151461082888709936, −4.38076422280720309572008084889, −3.87723769447527297907690780094, −2.93802890585523271826727935261, −2.55758750920185905468597115469, −1.96692575136634788700110646547, −1.17331170049337020311722581816, 0, 1.17331170049337020311722581816, 1.96692575136634788700110646547, 2.55758750920185905468597115469, 2.93802890585523271826727935261, 3.87723769447527297907690780094, 4.38076422280720309572008084889, 4.91465544529151461082888709936, 5.30581700616829522450723512615, 6.07765952753275119236893797977, 6.36438060854535008496393142433, 6.80985712973342974042195712008, 7.30258037467520962079345595243, 7.67181221204779541731261353120, 8.107051730609216110833704749959

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