Properties

Label 4-38e2-1.1-c1e2-0-2
Degree $4$
Conductor $1444$
Sign $1$
Analytic cond. $0.0920706$
Root an. cond. $0.550846$
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  + 2-s − 3-s − 6-s − 8·7-s − 8-s + 3·9-s + 6·11-s − 2·13-s − 8·14-s − 16-s + 6·17-s + 3·18-s − 7·19-s + 8·21-s + 6·22-s + 6·23-s + 24-s + 5·25-s − 2·26-s − 8·27-s + 4·31-s − 6·33-s + 6·34-s − 20·37-s − 7·38-s + 2·39-s − 9·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.408·6-s − 3.02·7-s − 0.353·8-s + 9-s + 1.80·11-s − 0.554·13-s − 2.13·14-s − 1/4·16-s + 1.45·17-s + 0.707·18-s − 1.60·19-s + 1.74·21-s + 1.27·22-s + 1.25·23-s + 0.204·24-s + 25-s − 0.392·26-s − 1.53·27-s + 0.718·31-s − 1.04·33-s + 1.02·34-s − 3.28·37-s − 1.13·38-s + 0.320·39-s − 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.0920706\)
Root analytic conductor: \(0.550846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1444,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5906645878\)
\(L(\frac12)\) \(\approx\) \(0.5906645878\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
19$C_2$ \( 1 + 7 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−16.56755263055327096788174289875, −16.22137496798641026167508889655, −15.28211244087574746901138481897, −15.20908919268970739743749330527, −14.16375082548199138628233952415, −13.66685992235296755885443165749, −12.73681987203851920762129508131, −12.63506419650880082403749061712, −12.32905909770147358575041978115, −11.45043803885142720835005436522, −10.15936809738882263373537801753, −10.12984738363830859482615609411, −9.299207338925210579842697562969, −8.779753188290020290978802133914, −6.90757797704302146249138961911, −6.83204172773892524372973604308, −6.22286096753511853920352949441, −5.16857224088052081785291055530, −3.82596749090966880616078458570, −3.35984729169594061369028138786, 3.35984729169594061369028138786, 3.82596749090966880616078458570, 5.16857224088052081785291055530, 6.22286096753511853920352949441, 6.83204172773892524372973604308, 6.90757797704302146249138961911, 8.779753188290020290978802133914, 9.299207338925210579842697562969, 10.12984738363830859482615609411, 10.15936809738882263373537801753, 11.45043803885142720835005436522, 12.32905909770147358575041978115, 12.63506419650880082403749061712, 12.73681987203851920762129508131, 13.66685992235296755885443165749, 14.16375082548199138628233952415, 15.20908919268970739743749330527, 15.28211244087574746901138481897, 16.22137496798641026167508889655, 16.56755263055327096788174289875

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