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Properties

Label 4-1218e2-1.1-c1e2-0-5
Degree $4$
Conductor $1483524$
Sign $1$
Analytic cond. $94.5907$
Root an. cond. $3.11861$
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 − 4·7-s − 9-s + 6·11-s + 16-s + 8·23-s − 4·25-s − 4·28-s − 2·29-s − 36-s + 10·37-s − 12·43-s + 6·44-s + 9·49-s + 6·53-s + 4·63-s + 64-s + 2·67-s + 12·71-s − 24·77-s + 10·79-s + 81-s + 8·92-s − 6·99-s − 4·100-s + 2·107-s − 6·109-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.51·7-s − 1/3·9-s + 1.80·11-s + 1/4·16-s + 1.66·23-s − 4/5·25-s − 0.755·28-s − 0.371·29-s − 1/6·36-s + 1.64·37-s − 1.82·43-s + 0.904·44-s + 9/7·49-s + 0.824·53-s + 0.503·63-s + 1/8·64-s + 0.244·67-s + 1.42·71-s − 2.73·77-s + 1.12·79-s + 1/9·81-s + 0.834·92-s − 0.603·99-s − 2/5·100-s + 0.193·107-s − 0.574·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1483524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1483524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1483524\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(94.5907\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1483524,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.141082513\)
\(L(\frac12)\) \(\approx\) \(2.141082513\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ag_w
13$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.13.a_m
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.23.ai_cg
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.31.a_w
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.37.ak_du
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.41.a_acs
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ag_dm
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.59.a_adc
61$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.61.a_acs
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.ac_ew
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.am_fm
73$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.73.a_k
79$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.79.ak_ha
83$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.83.a_abs
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.89.a_be
97$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.97.a_cc
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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

−8.000967253829813500558446674934, −7.23460286597369024037233102443, −6.92302162695566820039146940044, −6.70521424026297457938341323217, −6.18414529561532788664719652182, −5.99137916630712163634316645853, −5.37025166487349798978110893426, −4.82033387822604228960563601050, −4.18485176160201415634557113510, −3.59392455453036784641229923005, −3.45400530437571240578202008023, −2.79328966233465089533210340717, −2.22719295367084025026081089160, −1.40188382486373591922593464251, −0.65942048063755908033595792880, 0.65942048063755908033595792880, 1.40188382486373591922593464251, 2.22719295367084025026081089160, 2.79328966233465089533210340717, 3.45400530437571240578202008023, 3.59392455453036784641229923005, 4.18485176160201415634557113510, 4.82033387822604228960563601050, 5.37025166487349798978110893426, 5.99137916630712163634316645853, 6.18414529561532788664719652182, 6.70521424026297457938341323217, 6.92302162695566820039146940044, 7.23460286597369024037233102443, 8.000967253829813500558446674934

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