Properties

Label 4-490e2-1.1-c3e2-0-7
Degree $4$
Conductor $240100$
Sign $1$
Analytic cond. $835.842$
Root an. cond. $5.37688$
Motivic weight $3$
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  − 4·4-s − 20·5-s + 5·9-s − 74·11-s + 16·16-s − 216·19-s + 80·20-s + 275·25-s + 498·29-s + 268·31-s − 20·36-s − 412·41-s + 296·44-s − 100·45-s + 1.48e3·55-s − 4·59-s + 1.88e3·61-s − 64·64-s + 912·71-s + 864·76-s + 2.47e3·79-s − 320·80-s − 704·81-s − 440·89-s + 4.32e3·95-s − 370·99-s − 1.10e3·100-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s + 5/27·9-s − 2.02·11-s + 1/4·16-s − 2.60·19-s + 0.894·20-s + 11/5·25-s + 3.18·29-s + 1.55·31-s − 0.0925·36-s − 1.56·41-s + 1.01·44-s − 0.331·45-s + 3.62·55-s − 0.00882·59-s + 3.94·61-s − 1/8·64-s + 1.52·71-s + 1.30·76-s + 3.52·79-s − 0.447·80-s − 0.965·81-s − 0.524·89-s + 4.66·95-s − 0.375·99-s − 1.09·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(240100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(835.842\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{490} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 240100,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5754297499\)
\(L(\frac12)\) \(\approx\) \(0.5754297499\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{2} T^{2} \)
5$C_2$ \( 1 + 4 p T + p^{3} T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 37 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 1793 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 8145 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 108 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 19434 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 249 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 134 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 10250 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 206 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 17638 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 125277 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 297718 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 2 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 940 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 590290 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 456 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 355534 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 1239 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 960390 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 220 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 712321 T^{2} + p^{6} T^{4} \)
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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

−10.88984858452741850413261644410, −10.39147450357880000793897853948, −10.06534631538059304254435294140, −9.533275396470327408956705544434, −8.455793591411756842923583508376, −8.390646226696373737197570646662, −8.242133549592249647955203628102, −7.981984847407068270269091721237, −7.03965472727530106930365939734, −6.64487809452868843139584167533, −6.44901021091649594988061890626, −5.19701279982007051140619635344, −5.12336014758603854763038433786, −4.43398132836421108305922104039, −4.14376528456115719513893016371, −3.50332081764238534199157002609, −2.62107990296773824875865032064, −2.45957747222641861007880517789, −0.936028658556281577130460374182, −0.30722995468467903149198132616, 0.30722995468467903149198132616, 0.936028658556281577130460374182, 2.45957747222641861007880517789, 2.62107990296773824875865032064, 3.50332081764238534199157002609, 4.14376528456115719513893016371, 4.43398132836421108305922104039, 5.12336014758603854763038433786, 5.19701279982007051140619635344, 6.44901021091649594988061890626, 6.64487809452868843139584167533, 7.03965472727530106930365939734, 7.981984847407068270269091721237, 8.242133549592249647955203628102, 8.390646226696373737197570646662, 8.455793591411756842923583508376, 9.533275396470327408956705544434, 10.06534631538059304254435294140, 10.39147450357880000793897853948, 10.88984858452741850413261644410

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