Properties

Degree $4$
Conductor $100$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·4-s + 110·5-s + 290·9-s − 296·11-s + 256·16-s − 4.44e3·19-s − 1.76e3·20-s + 8.97e3·25-s + 540·29-s − 4.09e3·31-s − 4.64e3·36-s − 4.79e3·41-s + 4.73e3·44-s + 3.19e4·45-s + 8.65e3·49-s − 3.25e4·55-s + 7.94e4·59-s − 8.45e4·61-s − 4.09e3·64-s − 8.49e3·71-s + 7.10e4·76-s − 7.05e4·79-s + 2.81e4·80-s + 2.50e4·81-s + 1.70e5·89-s − 4.88e5·95-s − 8.58e4·99-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.96·5-s + 1.19·9-s − 0.737·11-s + 1/4·16-s − 2.82·19-s − 0.983·20-s + 2.87·25-s + 0.119·29-s − 0.765·31-s − 0.596·36-s − 0.445·41-s + 0.368·44-s + 2.34·45-s + 0.514·49-s − 1.45·55-s + 2.97·59-s − 2.91·61-s − 1/8·64-s − 0.200·71-s + 1.41·76-s − 1.27·79-s + 0.491·80-s + 0.424·81-s + 2.28·89-s − 5.55·95-s − 0.880·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{10} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 100,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.53353\)
\(L(\frac12)\) \(\approx\) \(1.53353\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{4} T^{2} \)
5$C_2$ \( 1 - 22 p T + p^{5} T^{2} \)
good3$C_2^2$ \( 1 - 290 T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 8650 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 + 148 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 274730 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 + 1354590 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 + 2220 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 11320170 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 270 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2048 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 119573530 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 2398 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 288754450 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 344584890 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 827605690 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 39740 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 42298 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 1669968610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 4248 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3239892370 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 35280 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7103795010 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 85210 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 7720618690 T^{2} + p^{10} T^{4} \)
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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

−20.63631268222401779868978150617, −19.43513243283377244668263818845, −18.56355373914592847949389886251, −18.35908324857109319320129519694, −17.42858007770352505493729629267, −17.05850236308791270866577775569, −16.18098083914274725706747896861, −15.05619563843942359927211657003, −14.50472295777770541541737003745, −13.40296404236440758697169464654, −13.11028404795236176114711471885, −12.52318821463482741978161202808, −10.57890465133649213233181899428, −10.36555378505865412595578756940, −9.379264379403550435948853783827, −8.514372490718778541168876741302, −6.89034138045290143199524724162, −5.86300359197973539959649504789, −4.58996945625990890057076318897, −2.02947078065342118326702508644, 2.02947078065342118326702508644, 4.58996945625990890057076318897, 5.86300359197973539959649504789, 6.89034138045290143199524724162, 8.514372490718778541168876741302, 9.379264379403550435948853783827, 10.36555378505865412595578756940, 10.57890465133649213233181899428, 12.52318821463482741978161202808, 13.11028404795236176114711471885, 13.40296404236440758697169464654, 14.50472295777770541541737003745, 15.05619563843942359927211657003, 16.18098083914274725706747896861, 17.05850236308791270866577775569, 17.42858007770352505493729629267, 18.35908324857109319320129519694, 18.56355373914592847949389886251, 19.43513243283377244668263818845, 20.63631268222401779868978150617

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