Properties

Label 6-1850e3-1.1-c1e3-0-1
Degree $6$
Conductor $6331625000$
Sign $1$
Analytic cond. $3223.63$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·3-s + 6·4-s − 9·6-s + 4·7-s − 10·8-s + 9-s − 11-s + 18·12-s + 12·13-s − 12·14-s + 15·16-s − 17-s − 3·18-s + 3·19-s + 12·21-s + 3·22-s + 8·23-s − 30·24-s − 36·26-s − 8·27-s + 24·28-s − 2·29-s − 6·31-s − 21·32-s − 3·33-s + 3·34-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.73·3-s + 3·4-s − 3.67·6-s + 1.51·7-s − 3.53·8-s + 1/3·9-s − 0.301·11-s + 5.19·12-s + 3.32·13-s − 3.20·14-s + 15/4·16-s − 0.242·17-s − 0.707·18-s + 0.688·19-s + 2.61·21-s + 0.639·22-s + 1.66·23-s − 6.12·24-s − 7.06·26-s − 1.53·27-s + 4.53·28-s − 0.371·29-s − 1.07·31-s − 3.71·32-s − 0.522·33-s + 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 5^{6} \cdot 37^{3}\)
Sign: $1$
Analytic conductor: \(3223.63\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 5^{6} \cdot 37^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.915999273\)
\(L(\frac12)\) \(\approx\) \(3.915999273\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
37$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 - p T + 8 T^{2} - 13 T^{3} + 8 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 4 T + 13 T^{2} - 22 T^{3} + 13 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + T + 10 T^{2} - 3 T^{3} + 10 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 12 T + 83 T^{2} - 358 T^{3} + 83 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + T + 30 T^{2} + 63 T^{3} + 30 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 3 T + 32 T^{2} - 35 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 8 T + 3 p T^{2} - 336 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 6 T + 75 T^{2} + 318 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 21 T + 254 T^{2} + 1937 T^{3} + 254 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 8 T + T^{2} + 388 T^{3} + p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 4 T + 93 T^{2} - 312 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 6 T + 59 T^{2} - 4 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 8 T + 49 T^{2} + 132 T^{3} + 49 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 8 T + 139 T^{2} + 686 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 13 T + 192 T^{2} - 25 p T^{3} + 192 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 8 T + 169 T^{2} + 846 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 31 T + 534 T^{2} - 5575 T^{3} + 534 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 22 T + 377 T^{2} - 3708 T^{3} + 377 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 7 T + 22 T^{2} + 657 T^{3} + 22 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 3 T + 240 T^{2} + 507 T^{3} + 240 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 6 T + 287 T^{2} - 1124 T^{3} + 287 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.307383448321807694381800379331, −8.069068671636531002664992776468, −7.965175745209693557237740538626, −7.85036462839381792939398339420, −7.16119790702190372030916596170, −7.09754322061188572025968622186, −6.86172017737696628188444217001, −6.39358377497674927360080064085, −6.36023965416904923034384284052, −5.77614325048529465289999736294, −5.48792846233368296067732696683, −5.36333268517365334261258117641, −5.09086812921316745902452633238, −4.53128558814086811483978676797, −3.89638786019622706403047285405, −3.86958488786682247800009026161, −3.37429030412132110413687281380, −3.12326921126962668452747384709, −3.10225356334277528534008017688, −2.40300268901560899349093291589, −2.02166940071156190352713018972, −1.86507172617046945991022350021, −1.49046591461553066379431683659, −0.872344792346611976073525024946, −0.75856436072018037282340273786, 0.75856436072018037282340273786, 0.872344792346611976073525024946, 1.49046591461553066379431683659, 1.86507172617046945991022350021, 2.02166940071156190352713018972, 2.40300268901560899349093291589, 3.10225356334277528534008017688, 3.12326921126962668452747384709, 3.37429030412132110413687281380, 3.86958488786682247800009026161, 3.89638786019622706403047285405, 4.53128558814086811483978676797, 5.09086812921316745902452633238, 5.36333268517365334261258117641, 5.48792846233368296067732696683, 5.77614325048529465289999736294, 6.36023965416904923034384284052, 6.39358377497674927360080064085, 6.86172017737696628188444217001, 7.09754322061188572025968622186, 7.16119790702190372030916596170, 7.85036462839381792939398339420, 7.965175745209693557237740538626, 8.069068671636531002664992776468, 8.307383448321807694381800379331

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