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Properties

Label 4-790272-1.1-c1e2-0-45
Degree $4$
Conductor $790272$
Sign $-1$
Analytic cond. $50.3884$
Root an. cond. $2.66429$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s + 7-s + 9-s + 4·11-s + 12·13-s + 38·25-s − 8·35-s − 8·43-s − 8·45-s − 24·47-s + 49-s − 32·55-s − 12·61-s + 63-s − 96·65-s − 16·67-s + 4·77-s + 81-s + 12·91-s + 4·99-s − 32·101-s + 32·103-s + 36·107-s + 20·113-s + 12·117-s − 10·121-s − 136·125-s + ⋯
L(s)  = 1  − 3.57·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 3.32·13-s + 38/5·25-s − 1.35·35-s − 1.21·43-s − 1.19·45-s − 3.50·47-s + 1/7·49-s − 4.31·55-s − 1.53·61-s + 0.125·63-s − 11.9·65-s − 1.95·67-s + 0.455·77-s + 1/9·81-s + 1.25·91-s + 0.402·99-s − 3.18·101-s + 3.15·103-s + 3.48·107-s + 1.88·113-s + 1.10·117-s − 0.909·121-s − 12.1·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(790272\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(50.3884\)
Root analytic conductor: \(2.66429\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 790272,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.5.i_ba
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.a_acc
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.a_o
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−8.077019225803717160884657215512, −7.60223474880564606977081998284, −7.38098041713554163025681861500, −6.71904731089565107044366408623, −6.34426698354997241941479748412, −6.06005190178770786910792401163, −4.87924854330324804890540113594, −4.63727069209135735412974054920, −4.16294073318355495294659943220, −3.65675458100353588864150742280, −3.37164547236104444198213794231, −3.30231377364451302978573657716, −1.46098811340915255187875146402, −1.14653761218391715748516515519, 0, 1.14653761218391715748516515519, 1.46098811340915255187875146402, 3.30231377364451302978573657716, 3.37164547236104444198213794231, 3.65675458100353588864150742280, 4.16294073318355495294659943220, 4.63727069209135735412974054920, 4.87924854330324804890540113594, 6.06005190178770786910792401163, 6.34426698354997241941479748412, 6.71904731089565107044366408623, 7.38098041713554163025681861500, 7.60223474880564606977081998284, 8.077019225803717160884657215512

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