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Properties

Label 4-1630475-1.1-c1e2-0-0
Degree $4$
Conductor $1630475$
Sign $1$
Analytic cond. $103.960$
Root an. cond. $3.19313$
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  + 4·4-s − 3·5-s − 5·9-s + 11-s + 12·16-s − 12·20-s + 4·25-s − 10·31-s − 20·36-s + 4·44-s + 15·45-s − 49-s − 3·55-s − 18·59-s + 32·64-s + 18·71-s − 36·80-s + 16·81-s + 6·89-s − 5·99-s + 16·100-s + 121-s − 40·124-s + 3·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 2·4-s − 1.34·5-s − 5/3·9-s + 0.301·11-s + 3·16-s − 2.68·20-s + 4/5·25-s − 1.79·31-s − 3.33·36-s + 0.603·44-s + 2.23·45-s − 1/7·49-s − 0.404·55-s − 2.34·59-s + 4·64-s + 2.13·71-s − 4.02·80-s + 16/9·81-s + 0.635·89-s − 0.502·99-s + 8/5·100-s + 1/11·121-s − 3.59·124-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1630475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1630475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1630475\)    =    \(5^{2} \cdot 7^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(103.960\)
Root analytic conductor: \(3.19313\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1630475,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.945996699\)
\(L(\frac12)\) \(\approx\) \(1.945996699\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5$C_2$ \( 1 + 3 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
11$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.2.a_ae
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.a_abv
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.59.s_hr
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.a_ef
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.71.as_ip
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.89.ag_hf
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.a_hl
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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

−7.78386483503909571561807910331, −7.52790988087429108012147116828, −7.02051968039498661407890301468, −6.59208147286700607708417702582, −6.26601684216824107557648129951, −5.74754794653349678875710884942, −5.43015975878698568723978852596, −4.89034994163855365043025916849, −4.05781061927516689834941885639, −3.59754168120135463214459998391, −3.18454604408396452211200185737, −2.85311216589406337019222737566, −2.15309334366858577702635245518, −1.63803594594217788940127740004, −0.55209935644750881682561305500, 0.55209935644750881682561305500, 1.63803594594217788940127740004, 2.15309334366858577702635245518, 2.85311216589406337019222737566, 3.18454604408396452211200185737, 3.59754168120135463214459998391, 4.05781061927516689834941885639, 4.89034994163855365043025916849, 5.43015975878698568723978852596, 5.74754794653349678875710884942, 6.26601684216824107557648129951, 6.59208147286700607708417702582, 7.02051968039498661407890301468, 7.52790988087429108012147116828, 7.78386483503909571561807910331

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