Properties

Label 4-676000-1.1-c1e2-0-40
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 − 5·13-s + 16-s + 3·17-s − 18-s + 20-s + 25-s − 5·26-s − 15·29-s + 32-s + 3·34-s − 36-s + 4·37-s + 40-s − 12·41-s − 45-s + 2·49-s + 50-s − 5·52-s + 6·53-s − 15·58-s − 7·61-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 − 1.38·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.223·20-s + 1/5·25-s − 0.980·26-s − 2.78·29-s + 0.176·32-s + 0.514·34-s − 1/6·36-s + 0.657·37-s + 0.158·40-s − 1.87·41-s − 0.149·45-s + 2/7·49-s + 0.141·50-s − 0.693·52-s + 0.824·53-s − 1.96·58-s − 0.896·61-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$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.11.a_i
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ad_bi
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.23.a_aq
29$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.p_ei
31$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.31.a_r
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.ae_bq
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.m_de
43$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.43.a_ck
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.ag_db
59$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.59.a_abu
61$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.h_ek
67$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \) 2.67.a_act
71$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \) 2.71.a_df
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.l_gs
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.83.a_acn
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−7.959209033692163976610246332262, −7.47060108935729935973996579032, −7.31128333405211587640243841465, −6.77776715040618879510484395967, −6.17866567754476012169503588827, −5.67753138713540389541205799663, −5.40513253680794061122459392960, −4.98812838771857430654371388464, −4.36770869706982745014302257945, −3.81835216360255949808626716767, −3.24991494949218000088237643501, −2.69907828760226865511504107018, −2.08052220191217320053681684302, −1.46756843426743641861341906563, 0, 1.46756843426743641861341906563, 2.08052220191217320053681684302, 2.69907828760226865511504107018, 3.24991494949218000088237643501, 3.81835216360255949808626716767, 4.36770869706982745014302257945, 4.98812838771857430654371388464, 5.40513253680794061122459392960, 5.67753138713540389541205799663, 6.17866567754476012169503588827, 6.77776715040618879510484395967, 7.31128333405211587640243841465, 7.47060108935729935973996579032, 7.959209033692163976610246332262

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