Properties

Label 4-70e2-1.1-c1e2-0-1
Degree $4$
Conductor $4900$
Sign $1$
Analytic cond. $0.312428$
Root an. cond. $0.747631$
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-s + 2·3-s + 5-s − 2·6-s − 4·7-s + 8-s + 3·9-s − 10-s − 3·11-s + 10·13-s + 4·14-s + 2·15-s − 16-s − 6·17-s − 3·18-s + 19-s − 8·21-s + 3·22-s − 3·23-s + 2·24-s − 10·26-s + 10·27-s − 12·29-s − 2·30-s + 4·31-s − 6·33-s + 6·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 0.447·5-s − 0.816·6-s − 1.51·7-s + 0.353·8-s + 9-s − 0.316·10-s − 0.904·11-s + 2.77·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.229·19-s − 1.74·21-s + 0.639·22-s − 0.625·23-s + 0.408·24-s − 1.96·26-s + 1.92·27-s − 2.22·29-s − 0.365·30-s + 0.718·31-s − 1.04·33-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.312428\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7714504273\)
\(L(\frac12)\) \(\approx\) \(0.7714504273\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−15.46884703286770629763712996237, −14.07200851584952180984754432976, −13.95269645933841741104857906200, −13.16981319807470047694460110414, −13.15044261811706142703383224756, −12.69204090063733291776161049702, −11.39122308004680081391904868716, −10.98886607033032661351288790856, −10.09854726735841160559946814581, −10.01262366676259156385878379647, −9.115226895573535331806530616824, −8.669190217470503777638136271838, −8.432608378056397329194640698368, −7.53085383801062044797924814498, −6.49009573887889420405119395625, −6.39989130578970503796910265395, −5.15545847896666123364690517603, −3.75548285742172031241020376086, −3.36443215474449269685995637012, −1.99133160960014135620033600093, 1.99133160960014135620033600093, 3.36443215474449269685995637012, 3.75548285742172031241020376086, 5.15545847896666123364690517603, 6.39989130578970503796910265395, 6.49009573887889420405119395625, 7.53085383801062044797924814498, 8.432608378056397329194640698368, 8.669190217470503777638136271838, 9.115226895573535331806530616824, 10.01262366676259156385878379647, 10.09854726735841160559946814581, 10.98886607033032661351288790856, 11.39122308004680081391904868716, 12.69204090063733291776161049702, 13.15044261811706142703383224756, 13.16981319807470047694460110414, 13.95269645933841741104857906200, 14.07200851584952180984754432976, 15.46884703286770629763712996237

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