Properties

Degree $4$
Conductor $100$
Sign $1$
Motivic weight $2$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 2·4-s + 8·6-s + 4·7-s + 8·9-s − 16·11-s − 8·12-s + 6·13-s − 8·14-s − 4·16-s + 14·17-s − 16·18-s − 16·21-s + 32·22-s − 4·23-s − 25·25-s − 12·26-s − 36·27-s + 8·28-s + 104·31-s + 8·32-s + 64·33-s − 28·34-s + 16·36-s − 6·37-s − 24·39-s + ⋯
L(s)  = 1  − 2-s − 4/3·3-s + 1/2·4-s + 4/3·6-s + 4/7·7-s + 8/9·9-s − 1.45·11-s − 2/3·12-s + 6/13·13-s − 4/7·14-s − 1/4·16-s + 0.823·17-s − 8/9·18-s − 0.761·21-s + 1.45·22-s − 0.173·23-s − 25-s − 0.461·26-s − 4/3·27-s + 2/7·28-s + 3.35·31-s + 1/4·32-s + 1.93·33-s − 0.823·34-s + 4/9·36-s − 0.162·37-s − 0.615·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

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

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.239370\)
\(L(\frac12)\) \(\approx\) \(0.239370\)
\(L(2)\) 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$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 + p^{2} T^{2} \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \)
19$C_2^2$ \( 1 - 322 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \)
31$C_2$ \( ( 1 - 52 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 84 T + 3528 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \)
59$C_2^2$ \( 1 - 6562 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 48 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 124 T + 7688 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 + 28 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2^2$ \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 9442 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.97670027424001189851460753414, −20.91840880760687132371699108974, −19.62802241243125925268870039984, −18.93596683106944575370378584993, −18.20751455061888268560137839128, −17.92223562712510451121164587598, −17.06911658410946393747976120432, −16.73674996682997615882681407582, −15.66615912972948546686921550101, −15.36823414016552186873536868229, −13.84524531625142941910940971889, −13.13024352722512341574554202219, −11.76761621055593255088125467995, −11.58377096028888216433546983152, −10.23281628919080119416287605008, −10.14654137527546697435584467874, −8.393364582816363596048703883492, −7.67558497356483516574264525363, −6.19183269878294484670114036539, −5.03540251964772219755791684917, 5.03540251964772219755791684917, 6.19183269878294484670114036539, 7.67558497356483516574264525363, 8.393364582816363596048703883492, 10.14654137527546697435584467874, 10.23281628919080119416287605008, 11.58377096028888216433546983152, 11.76761621055593255088125467995, 13.13024352722512341574554202219, 13.84524531625142941910940971889, 15.36823414016552186873536868229, 15.66615912972948546686921550101, 16.73674996682997615882681407582, 17.06911658410946393747976120432, 17.92223562712510451121164587598, 18.20751455061888268560137839128, 18.93596683106944575370378584993, 19.62802241243125925268870039984, 20.91840880760687132371699108974, 20.97670027424001189851460753414

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