Properties

Label 4-2850e2-1.1-c1e2-0-22
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 + 2·7-s + 4·8-s + 3·9-s + 4·11-s + 6·12-s + 2·13-s + 4·14-s + 5·16-s + 2·17-s + 6·18-s − 2·19-s + 4·21-s + 8·22-s + 2·23-s + 8·24-s + 4·26-s + 4·27-s + 6·28-s + 2·31-s + 6·32-s + 8·33-s + 4·34-s + 9·36-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s + 1.20·11-s + 1.73·12-s + 0.554·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s + 1.41·18-s − 0.458·19-s + 0.872·21-s + 1.70·22-s + 0.417·23-s + 1.63·24-s + 0.784·26-s + 0.769·27-s + 1.13·28-s + 0.359·31-s + 1.06·32-s + 1.39·33-s + 0.685·34-s + 3/2·36-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

\(L(1)\) \(\approx\) \(16.84828329\)
\(L(\frac12)\) \(\approx\) \(16.84828329\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 135 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 14 T + 164 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 135 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 167 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 56 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−8.726953230013388159642731996508, −8.703008140941807890927914800494, −8.002699271760752799235842594880, −7.965238822195098060537696902622, −7.26701332203973179580919135548, −7.23740121736467875167359994927, −6.56468310042259988921502428767, −6.30876323286851701940159826665, −5.86661213281215880643105074547, −5.56196184613748288934216639159, −4.76469398605454683043880058391, −4.64079110519782434084833201718, −4.28386169049636262686734589583, −3.69583606886267527919258304942, −3.52246917686238886789530786756, −3.03625380417388351953711543354, −2.31845545892331003034549717024, −2.21041592144861783304820456773, −1.22425598400350502537696339361, −1.20106690238198748164798732222, 1.20106690238198748164798732222, 1.22425598400350502537696339361, 2.21041592144861783304820456773, 2.31845545892331003034549717024, 3.03625380417388351953711543354, 3.52246917686238886789530786756, 3.69583606886267527919258304942, 4.28386169049636262686734589583, 4.64079110519782434084833201718, 4.76469398605454683043880058391, 5.56196184613748288934216639159, 5.86661213281215880643105074547, 6.30876323286851701940159826665, 6.56468310042259988921502428767, 7.23740121736467875167359994927, 7.26701332203973179580919135548, 7.965238822195098060537696902622, 8.002699271760752799235842594880, 8.703008140941807890927914800494, 8.726953230013388159642731996508

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