Properties

Label 4-570e2-1.1-c1e2-0-5
Degree $4$
Conductor $324900$
Sign $1$
Analytic cond. $20.7159$
Root an. cond. $2.13341$
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  − 2-s − 3-s + 5-s + 6-s − 2·7-s + 8-s − 10-s + 4·11-s + 3·13-s + 2·14-s − 15-s − 16-s − 4·17-s + 8·19-s + 2·21-s − 4·22-s + 6·23-s − 24-s − 3·26-s + 27-s + 10·29-s + 30-s + 2·31-s − 4·33-s + 4·34-s − 2·35-s − 10·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.832·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 1.83·19-s + 0.436·21-s − 0.852·22-s + 1.25·23-s − 0.204·24-s − 0.588·26-s + 0.192·27-s + 1.85·29-s + 0.182·30-s + 0.359·31-s − 0.696·33-s + 0.685·34-s − 0.338·35-s − 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(20.7159\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.119871739\)
\(L(\frac12)\) \(\approx\) \(1.119871739\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−10.82968777311078803666024264811, −10.49141368337666326423582416115, −9.916436371647121268699902052222, −9.795021897069123410852039953444, −9.078593819957897214792129023348, −8.840088507065860518693972175046, −8.655238444588747556259208083972, −7.895621802286099964824644110513, −7.22744542919858744365105500972, −6.88981659299806373152911804805, −6.47373468576235738619693083457, −6.10634893588634323077240572864, −5.52483200491050289155150484852, −4.83999106076969052721781158083, −4.57929764095546905524834171997, −3.43438431865943422985268001112, −3.42636963520616200921620401874, −2.40450431160513422748603935096, −1.37480648026459237086670625954, −0.809873558801618410664511583089, 0.809873558801618410664511583089, 1.37480648026459237086670625954, 2.40450431160513422748603935096, 3.42636963520616200921620401874, 3.43438431865943422985268001112, 4.57929764095546905524834171997, 4.83999106076969052721781158083, 5.52483200491050289155150484852, 6.10634893588634323077240572864, 6.47373468576235738619693083457, 6.88981659299806373152911804805, 7.22744542919858744365105500972, 7.895621802286099964824644110513, 8.655238444588747556259208083972, 8.840088507065860518693972175046, 9.078593819957897214792129023348, 9.795021897069123410852039953444, 9.916436371647121268699902052222, 10.49141368337666326423582416115, 10.82968777311078803666024264811

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