Properties

Label 4-570e2-1.1-c1e2-0-2
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  − 4-s − 2·5-s − 9-s − 8·11-s + 16-s + 2·19-s + 2·20-s − 25-s + 4·29-s − 12·31-s + 36-s + 8·44-s + 2·45-s + 14·49-s + 16·55-s − 4·59-s − 12·61-s − 64-s − 24·71-s − 2·76-s + 28·79-s − 2·80-s + 81-s − 8·89-s − 4·95-s + 8·99-s + 100-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 0.458·19-s + 0.447·20-s − 1/5·25-s + 0.742·29-s − 2.15·31-s + 1/6·36-s + 1.20·44-s + 0.298·45-s + 2·49-s + 2.15·55-s − 0.520·59-s − 1.53·61-s − 1/8·64-s − 2.84·71-s − 0.229·76-s + 3.15·79-s − 0.223·80-s + 1/9·81-s − 0.847·89-s − 0.410·95-s + 0.804·99-s + 1/10·100-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

\(L(1)\) \(\approx\) \(0.4506581117\)
\(L(\frac12)\) \(\approx\) \(0.4506581117\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−10.76378627726636236112099785040, −10.46462821165226532758204302613, −10.42780428767633370412130043710, −9.517752060107356672020385457221, −9.210128340046148514239332393854, −8.766363184929676765386913323261, −8.106015319848450357798688585191, −7.930333423499553177034582419579, −7.49629294753819375148836117429, −7.20021440091719279360429174353, −6.40534708988336544380642055374, −5.68731223812146267683738564969, −5.34485935136836988743198872688, −5.07455768381776820349955560035, −4.22696197960079737462113218221, −3.93427986242902300787553498051, −2.96184052688672467759281624440, −2.83488330187070679192426662326, −1.79229596142358938527930914928, −0.37345075326660318719524411743, 0.37345075326660318719524411743, 1.79229596142358938527930914928, 2.83488330187070679192426662326, 2.96184052688672467759281624440, 3.93427986242902300787553498051, 4.22696197960079737462113218221, 5.07455768381776820349955560035, 5.34485935136836988743198872688, 5.68731223812146267683738564969, 6.40534708988336544380642055374, 7.20021440091719279360429174353, 7.49629294753819375148836117429, 7.930333423499553177034582419579, 8.106015319848450357798688585191, 8.766363184929676765386913323261, 9.210128340046148514239332393854, 9.517752060107356672020385457221, 10.42780428767633370412130043710, 10.46462821165226532758204302613, 10.76378627726636236112099785040

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