Properties

Label 4-350e2-1.1-c1e2-0-2
Degree $4$
Conductor $122500$
Sign $1$
Analytic cond. $7.81070$
Root an. cond. $1.67175$
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 − 3·9-s − 10·11-s + 16-s + 6·19-s + 12·29-s − 8·31-s + 3·36-s + 22·41-s + 10·44-s − 49-s − 8·59-s − 4·61-s − 64-s − 20·71-s − 6·76-s + 4·79-s + 22·89-s + 30·99-s + 36·109-s − 12·116-s + 53·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 9-s − 3.01·11-s + 1/4·16-s + 1.37·19-s + 2.22·29-s − 1.43·31-s + 1/2·36-s + 3.43·41-s + 1.50·44-s − 1/7·49-s − 1.04·59-s − 0.512·61-s − 1/8·64-s − 2.37·71-s − 0.688·76-s + 0.450·79-s + 2.33·89-s + 3.01·99-s + 3.44·109-s − 1.11·116-s + 4.81·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(122500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(7.81070\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{350} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8408229588\)
\(L(\frac12)\) \(\approx\) \(0.8408229588\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−11.71213532525521739404529331675, −11.03248914126200209180492012179, −10.85847408081630898113138468309, −10.35013153478523174428802262470, −9.994343734392532640515735584001, −9.370667495287345675425294175021, −8.921340809713994409888151596341, −8.432901335985605660532466935114, −7.900121616147704622314811869964, −7.51444651436689669043190127183, −7.34160332609263357086643054592, −6.00923260737645566883775136857, −5.95850393584290940415092698715, −5.26795596119065845092726418093, −4.89340268853846218069727323280, −4.35538657252140008829443741148, −3.09017020285620483598189848717, −3.01173303664762003186045602547, −2.25592963732092920140091820731, −0.62531841110652152098686894891, 0.62531841110652152098686894891, 2.25592963732092920140091820731, 3.01173303664762003186045602547, 3.09017020285620483598189848717, 4.35538657252140008829443741148, 4.89340268853846218069727323280, 5.26795596119065845092726418093, 5.95850393584290940415092698715, 6.00923260737645566883775136857, 7.34160332609263357086643054592, 7.51444651436689669043190127183, 7.900121616147704622314811869964, 8.432901335985605660532466935114, 8.921340809713994409888151596341, 9.370667495287345675425294175021, 9.994343734392532640515735584001, 10.35013153478523174428802262470, 10.85847408081630898113138468309, 11.03248914126200209180492012179, 11.71213532525521739404529331675

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