Properties

Label 4-7098e2-1.1-c1e2-0-5
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
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  + 2·2-s − 2·3-s + 3·4-s + 3·5-s − 4·6-s + 2·7-s + 4·8-s + 3·9-s + 6·10-s − 7·11-s − 6·12-s + 4·14-s − 6·15-s + 5·16-s + 17-s + 6·18-s + 19-s + 9·20-s − 4·21-s − 14·22-s + 23-s − 8·24-s + 25-s − 4·27-s + 6·28-s − 29-s − 12·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.34·5-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s + 1.89·10-s − 2.11·11-s − 1.73·12-s + 1.06·14-s − 1.54·15-s + 5/4·16-s + 0.242·17-s + 1.41·18-s + 0.229·19-s + 2.01·20-s − 0.872·21-s − 2.98·22-s + 0.208·23-s − 1.63·24-s + 1/5·25-s − 0.769·27-s + 1.13·28-s − 0.185·29-s − 2.19·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.739130003\)
\(L(\frac12)\) \(\approx\) \(8.739130003\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 17 T + 154 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - T + 118 T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 19 T + 232 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−7.79571109458066037838248426877, −7.59365591100043862340432917494, −7.22669544383508386576816102649, −7.16479733070261756040025378850, −6.35692248005960227911859256725, −6.22866384066165521370774853818, −5.80737963788340633664991958501, −5.56180413767529545714201956514, −5.33917014821716389344046471942, −5.09766888889745016499914593069, −4.74737368744550953168740472843, −4.30527091043377436630949872107, −3.91593835201335488588665237966, −3.49679758868648860682835699441, −2.66110029507046352991247456887, −2.64886199597523044245473640126, −2.00123825167660119957600993247, −1.95860051293225030554226611163, −0.956285702528353515601824138628, −0.68938717688454125567124135762, 0.68938717688454125567124135762, 0.956285702528353515601824138628, 1.95860051293225030554226611163, 2.00123825167660119957600993247, 2.64886199597523044245473640126, 2.66110029507046352991247456887, 3.49679758868648860682835699441, 3.91593835201335488588665237966, 4.30527091043377436630949872107, 4.74737368744550953168740472843, 5.09766888889745016499914593069, 5.33917014821716389344046471942, 5.56180413767529545714201956514, 5.80737963788340633664991958501, 6.22866384066165521370774853818, 6.35692248005960227911859256725, 7.16479733070261756040025378850, 7.22669544383508386576816102649, 7.59365591100043862340432917494, 7.79571109458066037838248426877

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