Properties

Label 4-3450e2-1.1-c1e2-0-3
Degree $4$
Conductor $11902500$
Sign $1$
Analytic cond. $758.913$
Root an. cond. $5.24865$
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 − 9-s + 4·11-s + 16-s − 8·19-s − 16·31-s + 36-s − 4·41-s − 4·44-s + 14·49-s − 4·61-s − 64-s − 20·71-s + 8·76-s + 81-s − 8·89-s − 4·99-s − 4·109-s − 10·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1.20·11-s + 1/4·16-s − 1.83·19-s − 2.87·31-s + 1/6·36-s − 0.624·41-s − 0.603·44-s + 2·49-s − 0.512·61-s − 1/8·64-s − 2.37·71-s + 0.917·76-s + 1/9·81-s − 0.847·89-s − 0.402·99-s − 0.383·109-s − 0.909·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11902500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(758.913\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11902500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5277375936\)
\(L(\frac12)\) \(\approx\) \(0.5277375936\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−8.800841106865568606248544608588, −8.671708303761309882348714750303, −8.078056624496485883520350704103, −7.57687396507234789301980145130, −7.40359581359802599289776427613, −6.82413785473615188906291682597, −6.63068410472462867595353760249, −6.07897520738141816143513653911, −5.83156591356276053829149837625, −5.38805820420280999529783973466, −5.03493033654997584626407117887, −4.31560799559786495352568316536, −4.24475847355889004858784986094, −3.67155124745838427436242287520, −3.53292860288006180194309143443, −2.69948330882332794981554872726, −2.30796909921570149753098045218, −1.64447752652188570346616505164, −1.30215581999366280544888684773, −0.22235180309532984340696959816, 0.22235180309532984340696959816, 1.30215581999366280544888684773, 1.64447752652188570346616505164, 2.30796909921570149753098045218, 2.69948330882332794981554872726, 3.53292860288006180194309143443, 3.67155124745838427436242287520, 4.24475847355889004858784986094, 4.31560799559786495352568316536, 5.03493033654997584626407117887, 5.38805820420280999529783973466, 5.83156591356276053829149837625, 6.07897520738141816143513653911, 6.63068410472462867595353760249, 6.82413785473615188906291682597, 7.40359581359802599289776427613, 7.57687396507234789301980145130, 8.078056624496485883520350704103, 8.671708303761309882348714750303, 8.800841106865568606248544608588

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