Properties

Label 4-8470e2-1.1-c1e2-0-3
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 3·4-s − 2·5-s − 2·6-s + 2·7-s − 4·8-s − 4·9-s + 4·10-s + 3·12-s − 6·13-s − 4·14-s − 2·15-s + 5·16-s + 3·17-s + 8·18-s + 9·19-s − 6·20-s + 2·21-s + 8·23-s − 4·24-s + 3·25-s + 12·26-s − 6·27-s + 6·28-s + 4·30-s − 4·31-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.894·5-s − 0.816·6-s + 0.755·7-s − 1.41·8-s − 4/3·9-s + 1.26·10-s + 0.866·12-s − 1.66·13-s − 1.06·14-s − 0.516·15-s + 5/4·16-s + 0.727·17-s + 1.88·18-s + 2.06·19-s − 1.34·20-s + 0.436·21-s + 1.66·23-s − 0.816·24-s + 3/5·25-s + 2.35·26-s − 1.15·27-s + 1.13·28-s + 0.730·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.411088748\)
\(L(\frac12)\) \(\approx\) \(1.411088748\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 18 T + 150 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + T + 71 T^{2} + p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 15 T + 163 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 9 T + 153 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T + 135 T^{2} + p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 11 T + 185 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 23 T + 299 T^{2} + 23 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 19 T + 283 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
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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

−8.011769568777500725586441801606, −7.69873217908466397607728991657, −7.25553467154901869464876109397, −7.23722335382240647760572628652, −6.90223579070844794460383068102, −6.61277221642534947516244448214, −5.54754371918648543913602314947, −5.52951114911103229183786782461, −5.33414615019700093194926103879, −5.15869109684819681564229747236, −4.25350862504594967747850334154, −4.07692888292210612502654930023, −3.32254418999150443307619818676, −3.17671905048671629567339581120, −2.71662742058705881527167524169, −2.62881031823272312295719132076, −1.78781539105826097738640771409, −1.55111260496128532395781375442, −0.67328932883577120246152414740, −0.51442012573373620441364043649, 0.51442012573373620441364043649, 0.67328932883577120246152414740, 1.55111260496128532395781375442, 1.78781539105826097738640771409, 2.62881031823272312295719132076, 2.71662742058705881527167524169, 3.17671905048671629567339581120, 3.32254418999150443307619818676, 4.07692888292210612502654930023, 4.25350862504594967747850334154, 5.15869109684819681564229747236, 5.33414615019700093194926103879, 5.52951114911103229183786782461, 5.54754371918648543913602314947, 6.61277221642534947516244448214, 6.90223579070844794460383068102, 7.23722335382240647760572628652, 7.25553467154901869464876109397, 7.69873217908466397607728991657, 8.011769568777500725586441801606

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