Properties

Label 4-3850e2-1.1-c1e2-0-26
Degree $4$
Conductor $14822500$
Sign $1$
Analytic cond. $945.095$
Root an. cond. $5.54458$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·9-s − 2·11-s + 16-s − 12·19-s − 8·29-s − 2·36-s + 2·44-s − 49-s + 4·61-s − 64-s − 24·71-s + 12·76-s − 20·79-s − 5·81-s − 28·89-s − 4·99-s + 20·101-s − 8·109-s + 8·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s + 2/3·9-s − 0.603·11-s + 1/4·16-s − 2.75·19-s − 1.48·29-s − 1/3·36-s + 0.301·44-s − 1/7·49-s + 0.512·61-s − 1/8·64-s − 2.84·71-s + 1.37·76-s − 2.25·79-s − 5/9·81-s − 2.96·89-s − 0.402·99-s + 1.99·101-s − 0.766·109-s + 0.742·116-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14822500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(945.095\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 14822500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \)
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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

−8.475614877622356905547590549355, −8.085814314957339577995512127655, −7.48377765896644411927321248435, −7.16432260472826718373807292065, −7.02161842990201074160993018967, −6.40024451779496509418977018619, −6.00022702765669506663197129910, −5.77463598435391426678897691443, −5.33895515468417569063026828636, −4.75221247308167672168137178437, −4.46776983012529208409487583369, −4.05551786322025377153957419828, −3.93373367022926308644368475538, −3.15298525866320766713267640342, −2.76009765824846737722586717662, −2.16493235192889374264827673213, −1.76191833197185291318646263585, −1.22204359103769649818715128951, 0, 0, 1.22204359103769649818715128951, 1.76191833197185291318646263585, 2.16493235192889374264827673213, 2.76009765824846737722586717662, 3.15298525866320766713267640342, 3.93373367022926308644368475538, 4.05551786322025377153957419828, 4.46776983012529208409487583369, 4.75221247308167672168137178437, 5.33895515468417569063026828636, 5.77463598435391426678897691443, 6.00022702765669506663197129910, 6.40024451779496509418977018619, 7.02161842990201074160993018967, 7.16432260472826718373807292065, 7.48377765896644411927321248435, 8.085814314957339577995512127655, 8.475614877622356905547590549355

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