Properties

Label 4-1560e2-1.1-c1e2-0-10
Degree $4$
Conductor $2433600$
Sign $1$
Analytic cond. $155.168$
Root an. cond. $3.52939$
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  − 2·5-s − 9-s + 10·11-s + 8·19-s − 25-s + 8·29-s + 22·41-s + 2·45-s − 11·49-s − 20·55-s − 24·59-s − 14·61-s − 14·71-s + 10·79-s + 81-s + 6·89-s − 16·95-s − 10·99-s − 12·101-s + 24·109-s + 53·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + ⋯
L(s)  = 1  − 0.894·5-s − 1/3·9-s + 3.01·11-s + 1.83·19-s − 1/5·25-s + 1.48·29-s + 3.43·41-s + 0.298·45-s − 1.57·49-s − 2.69·55-s − 3.12·59-s − 1.79·61-s − 1.66·71-s + 1.12·79-s + 1/9·81-s + 0.635·89-s − 1.64·95-s − 1.00·99-s − 1.19·101-s + 2.29·109-s + 4.81·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2433600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(155.168\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2433600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.606233462\)
\(L(\frac12)\) \(\approx\) \(2.606233462\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 193 T^{2} + p^{2} T^{4} \)
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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

−9.552244043282663860021503939118, −9.173634983474205144330880674826, −9.113948906419845761224020086676, −8.372128110346176017244995400548, −8.091572675578105192101626330397, −7.50242587802434220348209059757, −7.37614899805581717834938442773, −6.86762031436111040662013913849, −6.30143934535369195218255744222, −5.97634103260763198120458637157, −5.88390238191610028515611024548, −4.77058663860817632306777909911, −4.59839025359759732173204797102, −4.20144596758247429734718963107, −3.68305852272022394495996522875, −3.11582553573484230142700598824, −2.99063762398255402286193420137, −1.82083390214033712196011348908, −1.26926690588574235504681386438, −0.74037234441383962200328963926, 0.74037234441383962200328963926, 1.26926690588574235504681386438, 1.82083390214033712196011348908, 2.99063762398255402286193420137, 3.11582553573484230142700598824, 3.68305852272022394495996522875, 4.20144596758247429734718963107, 4.59839025359759732173204797102, 4.77058663860817632306777909911, 5.88390238191610028515611024548, 5.97634103260763198120458637157, 6.30143934535369195218255744222, 6.86762031436111040662013913849, 7.37614899805581717834938442773, 7.50242587802434220348209059757, 8.091572675578105192101626330397, 8.372128110346176017244995400548, 9.113948906419845761224020086676, 9.173634983474205144330880674826, 9.552244043282663860021503939118

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