Properties

Label 4-490e2-1.1-c1e2-0-14
Degree $4$
Conductor $240100$
Sign $1$
Analytic cond. $15.3089$
Root an. cond. $1.97804$
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 + 4·3-s + 3·4-s + 2·5-s − 8·6-s − 4·8-s + 8·9-s − 4·10-s + 4·11-s + 12·12-s − 4·13-s + 8·15-s + 5·16-s + 8·17-s − 16·18-s + 4·19-s + 6·20-s − 8·22-s − 8·23-s − 16·24-s + 3·25-s + 8·26-s + 12·27-s − 4·29-s − 16·30-s − 6·32-s + 16·33-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.30·3-s + 3/2·4-s + 0.894·5-s − 3.26·6-s − 1.41·8-s + 8/3·9-s − 1.26·10-s + 1.20·11-s + 3.46·12-s − 1.10·13-s + 2.06·15-s + 5/4·16-s + 1.94·17-s − 3.77·18-s + 0.917·19-s + 1.34·20-s − 1.70·22-s − 1.66·23-s − 3.26·24-s + 3/5·25-s + 1.56·26-s + 2.30·27-s − 0.742·29-s − 2.92·30-s − 1.06·32-s + 2.78·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(240100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(15.3089\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{490} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 240100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.690216343\)
\(L(\frac12)\) \(\approx\) \(2.690216343\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 20 T + 216 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 166 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 152 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
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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

−10.88459580007985610221619904271, −10.28971780065301490311421084737, −9.943249755000821756197060500778, −9.749815474932624267620391551470, −9.318640555922559278614043276088, −9.117572308403497337302090990962, −8.410575894350371954381270306616, −8.342847444009443390803294271408, −7.56099190688200919690633753526, −7.51000711774831488048519746298, −6.97136286400532135914052356201, −6.30460120710322372141500888161, −5.65977706098032818667730898633, −5.21403970872239700718804396046, −3.93324940979279260936738912402, −3.73518003555372199495859711444, −2.76040274213252714370841685164, −2.66773972862876738845706573219, −1.77576856829648359670824497850, −1.28269316591121626929788201140, 1.28269316591121626929788201140, 1.77576856829648359670824497850, 2.66773972862876738845706573219, 2.76040274213252714370841685164, 3.73518003555372199495859711444, 3.93324940979279260936738912402, 5.21403970872239700718804396046, 5.65977706098032818667730898633, 6.30460120710322372141500888161, 6.97136286400532135914052356201, 7.51000711774831488048519746298, 7.56099190688200919690633753526, 8.342847444009443390803294271408, 8.410575894350371954381270306616, 9.117572308403497337302090990962, 9.318640555922559278614043276088, 9.749815474932624267620391551470, 9.943249755000821756197060500778, 10.28971780065301490311421084737, 10.88459580007985610221619904271

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