Properties

Label 4-1344e2-1.1-c1e2-0-3
Degree $4$
Conductor $1806336$
Sign $1$
Analytic cond. $115.173$
Root an. cond. $3.27595$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 7-s + 3·11-s + 3·15-s + 6·17-s + 2·19-s + 21-s + 25-s + 27-s − 18·29-s + 5·31-s − 3·33-s + 3·35-s + 10·37-s + 12·47-s − 6·49-s − 6·51-s − 9·53-s − 9·55-s − 2·57-s + 9·59-s − 24·67-s − 12·73-s − 75-s − 3·77-s − 9·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.377·7-s + 0.904·11-s + 0.774·15-s + 1.45·17-s + 0.458·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 3.34·29-s + 0.898·31-s − 0.522·33-s + 0.507·35-s + 1.64·37-s + 1.75·47-s − 6/7·49-s − 0.840·51-s − 1.23·53-s − 1.21·55-s − 0.264·57-s + 1.17·59-s − 2.93·67-s − 1.40·73-s − 0.115·75-s − 0.341·77-s − 1.01·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(115.173\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8661096842\)
\(L(\frac12)\) \(\approx\) \(0.8661096842\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 24 T + 259 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T + 101 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−9.837784220773302797077001387848, −9.336637123082073213891484690161, −9.122798911171030368346284100504, −8.659877364889133030308287191103, −7.944140995474845982423108013042, −7.67116182839822955782882178504, −7.58247777544775078486011408921, −7.04089998824801707265326317631, −6.56038747061640353545031026253, −5.88940756302686263316643812330, −5.76802753435836959155136317734, −5.39653681122788288615210998714, −4.52618933989122503672964356492, −4.21359835520414926642259265583, −3.89062182515766854070625239209, −3.16168007392355848246001559615, −3.09362576197782670383268073944, −1.95381724152648213569233911448, −1.26347717082733020909776781681, −0.44094050816918797106739237742, 0.44094050816918797106739237742, 1.26347717082733020909776781681, 1.95381724152648213569233911448, 3.09362576197782670383268073944, 3.16168007392355848246001559615, 3.89062182515766854070625239209, 4.21359835520414926642259265583, 4.52618933989122503672964356492, 5.39653681122788288615210998714, 5.76802753435836959155136317734, 5.88940756302686263316643812330, 6.56038747061640353545031026253, 7.04089998824801707265326317631, 7.58247777544775078486011408921, 7.67116182839822955782882178504, 7.944140995474845982423108013042, 8.659877364889133030308287191103, 9.122798911171030368346284100504, 9.336637123082073213891484690161, 9.837784220773302797077001387848

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