Properties

Label 4-1305e2-1.1-c1e2-0-9
Degree $4$
Conductor $1703025$
Sign $1$
Analytic cond. $108.586$
Root an. cond. $3.22807$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 10·11-s + 12·16-s − 8·19-s − 5·25-s + 2·29-s − 4·31-s − 10·41-s + 40·44-s − 6·49-s − 20·59-s + 24·61-s + 32·64-s + 20·71-s − 32·76-s + 28·79-s + 20·89-s − 20·100-s − 30·101-s + 18·109-s + 8·116-s + 53·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2·4-s + 3.01·11-s + 3·16-s − 1.83·19-s − 25-s + 0.371·29-s − 0.718·31-s − 1.56·41-s + 6.03·44-s − 6/7·49-s − 2.60·59-s + 3.07·61-s + 4·64-s + 2.37·71-s − 3.67·76-s + 3.15·79-s + 2.11·89-s − 2·100-s − 2.98·101-s + 1.72·109-s + 0.742·116-s + 4.81·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1703025\)    =    \(3^{4} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(108.586\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1703025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.892487297\)
\(L(\frac12)\) \(\approx\) \(4.892487297\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_2$ \( 1 + p T^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 69 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 81 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 101 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 101 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 161 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 149 T^{2} + p^{2} T^{4} \)
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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

−9.727205680798607181460425532686, −9.589523649860288525635951763469, −9.109982897195062097089936369439, −8.450617382195320946767908395323, −8.302914218659634652233113526816, −7.81213255114626332034570237987, −7.09390712474675119723080961195, −6.98535843067089708997690899360, −6.41795676981730713935050219298, −6.25108361069847242003643152718, −6.18972116085907075969447335988, −5.29335058982311909944483536725, −4.79823609229910988442190421024, −3.97735186323785849554415579634, −3.61057718120716377329266955273, −3.52662449276075884672404318182, −2.50037320295235999844395912835, −1.91003404237697701351596757849, −1.74953945847471012593343290395, −0.943695838140188810001163655601, 0.943695838140188810001163655601, 1.74953945847471012593343290395, 1.91003404237697701351596757849, 2.50037320295235999844395912835, 3.52662449276075884672404318182, 3.61057718120716377329266955273, 3.97735186323785849554415579634, 4.79823609229910988442190421024, 5.29335058982311909944483536725, 6.18972116085907075969447335988, 6.25108361069847242003643152718, 6.41795676981730713935050219298, 6.98535843067089708997690899360, 7.09390712474675119723080961195, 7.81213255114626332034570237987, 8.302914218659634652233113526816, 8.450617382195320946767908395323, 9.109982897195062097089936369439, 9.589523649860288525635951763469, 9.727205680798607181460425532686

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