Properties

Label 4-676000-1.1-c1e2-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s + 5·9-s − 10-s + 2·13-s + 16-s − 3·17-s + 5·18-s − 20-s + 25-s + 2·26-s + 15·29-s + 32-s − 3·34-s + 5·36-s − 16·37-s − 40-s − 12·41-s − 5·45-s + 2·49-s + 50-s + 2·52-s + 3·53-s + 15·58-s + 17·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s + 5/3·9-s − 0.316·10-s + 0.554·13-s + 1/4·16-s − 0.727·17-s + 1.17·18-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 2.78·29-s + 0.176·32-s − 0.514·34-s + 5/6·36-s − 2.63·37-s − 0.158·40-s − 1.87·41-s − 0.745·45-s + 2/7·49-s + 0.141·50-s + 0.277·52-s + 0.412·53-s + 1.96·58-s + 2.17·61-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: \(0\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.523305643\)
\(L(\frac12)\) \(\approx\) \(3.523305643\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.23.a_ae
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.29.ap_ei
31$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.31.a_r
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.43.a_bd
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.47.a_e
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.ad_dk
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.61.ar_gs
67$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.67.a_abp
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.71.a_ck
73$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.b_fo
79$C_2^2$ \( 1 - 47 T^{2} + p^{2} T^{4} \) 2.79.a_abv
83$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.83.a_aeg
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.as_jq
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.97.at_ke
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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

−8.333373445526685847049430112725, −7.86260032159893838626284289910, −7.24437962994790367630257369499, −6.88538548366697790062017292372, −6.59275830021601663022409339195, −6.32301650866034743141117722267, −5.33062713635068312401889254402, −5.06652809662714451458358776841, −4.56619687315331338989964972468, −4.14004602379633686505785428295, −3.57765317833638405894155643696, −3.21374702541597366128391059213, −2.28385795121003706929046548096, −1.70856012192976828479497819710, −0.886936893152223500011099410066, 0.886936893152223500011099410066, 1.70856012192976828479497819710, 2.28385795121003706929046548096, 3.21374702541597366128391059213, 3.57765317833638405894155643696, 4.14004602379633686505785428295, 4.56619687315331338989964972468, 5.06652809662714451458358776841, 5.33062713635068312401889254402, 6.32301650866034743141117722267, 6.59275830021601663022409339195, 6.88538548366697790062017292372, 7.24437962994790367630257369499, 7.86260032159893838626284289910, 8.333373445526685847049430112725

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