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Properties

Label 4-819e2-1.1-c1e2-0-17
Degree $4$
Conductor $670761$
Sign $1$
Analytic cond. $42.7683$
Root an. cond. $2.55729$
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  + 4-s − 3·16-s + 8·19-s + 6·25-s − 8·31-s + 12·37-s − 7·49-s + 8·61-s − 7·64-s − 8·73-s + 8·76-s − 8·79-s + 32·97-s + 6·100-s + 16·103-s − 4·109-s − 6·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 12·148-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/2·4-s − 3/4·16-s + 1.83·19-s + 6/5·25-s − 1.43·31-s + 1.97·37-s − 49-s + 1.02·61-s − 7/8·64-s − 0.936·73-s + 0.917·76-s − 0.900·79-s + 3.24·97-s + 3/5·100-s + 1.57·103-s − 0.383·109-s − 0.545·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.986·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(670761\)    =    \(3^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(42.7683\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 670761,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.282184056\)
\(L(\frac12)\) \(\approx\) \(2.282184056\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
7$C_2$ \( 1 + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.19.ai_bm
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.am_dq
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.41.a_c
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.47.a_k
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.53.a_k
59$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.59.a_du
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.ai_bm
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.i_gc
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.i_gc
83$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.83.a_ack
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) 2.97.abg_re
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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

−8.399478269422491857933120887222, −7.69284780302732796081243896828, −7.40329455805969337586046385082, −7.17169795937423532047457496655, −6.52840917278858378508494049082, −6.12651454782128697564405302379, −5.65508043751951853527023430324, −5.04865422437471760867508906365, −4.73684859222634654452372157160, −4.08287439981682554370461249932, −3.38458740737852116554504178998, −2.98637874970414366677564138506, −2.36048640147549282635010547391, −1.62773208662784761853899794309, −0.77550722625892471951964410030, 0.77550722625892471951964410030, 1.62773208662784761853899794309, 2.36048640147549282635010547391, 2.98637874970414366677564138506, 3.38458740737852116554504178998, 4.08287439981682554370461249932, 4.73684859222634654452372157160, 5.04865422437471760867508906365, 5.65508043751951853527023430324, 6.12651454782128697564405302379, 6.52840917278858378508494049082, 7.17169795937423532047457496655, 7.40329455805969337586046385082, 7.69284780302732796081243896828, 8.399478269422491857933120887222

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