Properties

Label 4-2850e2-1.1-c1e2-0-16
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
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 − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 3·9-s + 2·11-s − 6·12-s + 4·13-s + 5·16-s + 8·17-s + 6·18-s − 2·19-s + 4·22-s + 2·23-s − 8·24-s + 8·26-s − 4·27-s − 14·29-s − 2·31-s + 6·32-s − 4·33-s + 16·34-s + 9·36-s + 20·37-s − 4·38-s − 8·39-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 1.41·8-s + 9-s + 0.603·11-s − 1.73·12-s + 1.10·13-s + 5/4·16-s + 1.94·17-s + 1.41·18-s − 0.458·19-s + 0.852·22-s + 0.417·23-s − 1.63·24-s + 1.56·26-s − 0.769·27-s − 2.59·29-s − 0.359·31-s + 1.06·32-s − 0.696·33-s + 2.74·34-s + 3/2·36-s + 3.28·37-s − 0.648·38-s − 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.233832647\)
\(L(\frac12)\) \(\approx\) \(6.233832647\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 14 T + 97 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 10 T + 121 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 2 T + 113 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 136 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 2 T + 119 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 139 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
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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

−8.923105543831959452592431231343, −8.591281669975302037044525835906, −8.032995874295812616320836664602, −7.51780013003489545750272305722, −7.35326246101385024710560693505, −7.10350260918887853504427632955, −6.29598148765238379900483252366, −6.16598670402954719517792420106, −5.89560895376279476522038578295, −5.59844681752436845088987230563, −5.16120985443499075813917329518, −4.74931527328241733480287833120, −4.20934265565401993689806227687, −3.93227443534041960253736021553, −3.41073646180013937204651051637, −3.28672443233261357972023514643, −2.32568856046296764835560675109, −1.89974976580143837146758565300, −1.17778454316745243393976271951, −0.77644326464205890941869909154, 0.77644326464205890941869909154, 1.17778454316745243393976271951, 1.89974976580143837146758565300, 2.32568856046296764835560675109, 3.28672443233261357972023514643, 3.41073646180013937204651051637, 3.93227443534041960253736021553, 4.20934265565401993689806227687, 4.74931527328241733480287833120, 5.16120985443499075813917329518, 5.59844681752436845088987230563, 5.89560895376279476522038578295, 6.16598670402954719517792420106, 6.29598148765238379900483252366, 7.10350260918887853504427632955, 7.35326246101385024710560693505, 7.51780013003489545750272305722, 8.032995874295812616320836664602, 8.591281669975302037044525835906, 8.923105543831959452592431231343

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