Properties

Label 4-6300e2-1.1-c1e2-0-3
Degree $4$
Conductor $39690000$
Sign $1$
Analytic cond. $2530.66$
Root an. cond. $7.09265$
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  − 4·11-s + 8·19-s − 4·29-s − 49-s − 16·59-s + 12·61-s − 28·71-s − 24·79-s − 32·101-s + 4·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 1.20·11-s + 1.83·19-s − 0.742·29-s − 1/7·49-s − 2.08·59-s + 1.53·61-s − 3.32·71-s − 2.70·79-s − 3.18·101-s + 0.383·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(39690000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2530.66\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 39690000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5043638897\)
\(L(\frac12)\) \(\approx\) \(0.5043638897\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.331916542968755470370379987275, −7.63254544422784445227909246295, −7.56156523973098523448654439858, −7.29927822443569644251915368845, −6.98257935910849707866313867302, −6.28503661962175324389174650315, −6.17524548117577484452384575213, −5.50892203225646806213486388173, −5.48520834236951990730073215459, −5.07829159415646891310801848433, −4.70227468616556131251141092364, −4.09716352807958767177191724275, −3.96974842120990090346244940649, −3.24612504888222370662574153703, −2.87599718117033692806891964981, −2.79409807269288405723341714778, −2.11402219007738449201948800389, −1.35607461386975175965578174381, −1.29062352290121839748081818876, −0.18044265537975429540259445565, 0.18044265537975429540259445565, 1.29062352290121839748081818876, 1.35607461386975175965578174381, 2.11402219007738449201948800389, 2.79409807269288405723341714778, 2.87599718117033692806891964981, 3.24612504888222370662574153703, 3.96974842120990090346244940649, 4.09716352807958767177191724275, 4.70227468616556131251141092364, 5.07829159415646891310801848433, 5.48520834236951990730073215459, 5.50892203225646806213486388173, 6.17524548117577484452384575213, 6.28503661962175324389174650315, 6.98257935910849707866313867302, 7.29927822443569644251915368845, 7.56156523973098523448654439858, 7.63254544422784445227909246295, 8.331916542968755470370379987275

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