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Properties

Label 4-819e2-1.1-c1e2-0-50
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 7-s − 13-s − 2·19-s − 2·25-s + 2·28-s − 4·31-s − 12·37-s + 5·43-s − 6·49-s − 2·52-s − 9·61-s − 8·64-s − 9·67-s − 16·73-s − 4·76-s − 10·79-s − 91-s − 5·97-s − 4·100-s + 6·103-s + 12·109-s + 5·121-s − 8·124-s + 127-s + 131-s − 2·133-s + ⋯
L(s)  = 1  + 4-s + 0.377·7-s − 0.277·13-s − 0.458·19-s − 2/5·25-s + 0.377·28-s − 0.718·31-s − 1.97·37-s + 0.762·43-s − 6/7·49-s − 0.277·52-s − 1.15·61-s − 64-s − 1.09·67-s − 1.87·73-s − 0.458·76-s − 1.12·79-s − 0.104·91-s − 0.507·97-s − 2/5·100-s + 0.591·103-s + 1.14·109-s + 5/11·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s − 0.173·133-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: \(1\)
Selberg data: \((4,\ 670761,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
13$C_2$ \( 1 + T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.17.a_bf
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.c_d
23$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.23.a_r
29$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.29.a_ab
31$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.e_cf
37$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.m_ef
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.41.a_acd
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.43.af_dm
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.47.a_c
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \) 2.59.a_abx
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.61.j_fg
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.j_fy
71$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \) 2.71.a_dn
73$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.q_ht
79$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.79.k_fr
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.89.a_bp
97$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.f_fo
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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

−8.050410832388563778283814904588, −7.51267285158613443437371192202, −7.22480369990723168175860097186, −6.93704357248659722619326490804, −6.26125532523106571783012657181, −5.92636681849482320525136210915, −5.49768116970693892151520196744, −4.72200552444619457588408530073, −4.52160749380226083190509803499, −3.71949575563839321065311570435, −3.17750376798647924400430682627, −2.59763781104362218613553900494, −1.89420878522561252854471951859, −1.51508341509677977074258029538, 0, 1.51508341509677977074258029538, 1.89420878522561252854471951859, 2.59763781104362218613553900494, 3.17750376798647924400430682627, 3.71949575563839321065311570435, 4.52160749380226083190509803499, 4.72200552444619457588408530073, 5.49768116970693892151520196744, 5.92636681849482320525136210915, 6.26125532523106571783012657181, 6.93704357248659722619326490804, 7.22480369990723168175860097186, 7.51267285158613443437371192202, 8.050410832388563778283814904588

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