Properties

Label 6-535e3-535.534-c0e3-0-0
Degree $6$
Conductor $153130375$
Sign $1$
Analytic cond. $0.0190341$
Root an. cond. $0.516720$
Motivic weight $0$
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  + 2-s − 3·5-s + 7-s + 3·9-s − 3·10-s − 11-s + 14-s + 17-s + 3·18-s − 19-s − 22-s + 6·25-s − 29-s + 34-s − 3·35-s − 38-s − 41-s + 43-s − 9·45-s + 6·50-s + 3·55-s − 58-s − 61-s + 3·63-s + 67-s − 3·70-s + 73-s + ⋯
L(s)  = 1  + 2-s − 3·5-s + 7-s + 3·9-s − 3·10-s − 11-s + 14-s + 17-s + 3·18-s − 19-s − 22-s + 6·25-s − 29-s + 34-s − 3·35-s − 38-s − 41-s + 43-s − 9·45-s + 6·50-s + 3·55-s − 58-s − 61-s + 3·63-s + 67-s − 3·70-s + 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 107^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 107^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(5^{3} \cdot 107^{3}\)
Sign: $1$
Analytic conductor: \(0.0190341\)
Root analytic conductor: \(0.516720\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{535} (534, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 5^{3} \cdot 107^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8138596026\)
\(L(\frac12)\) \(\approx\) \(0.8138596026\)
\(L(1)\) 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)$
bad5$C_1$ \( ( 1 + T )^{3} \)
107$C_1$ \( ( 1 + T )^{3} \)
good2$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
7$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
11$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
17$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
19$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
29$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
41$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
43$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
61$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
67$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
79$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
89$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
97$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
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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

−10.26982506078357238541825753478, −9.467994169589191855473599328590, −9.324141254380461169760587001755, −8.972750987382410267705541612372, −8.490467274549382096495902889743, −8.047008761313330853147939240620, −7.956946056206819531068158383000, −7.81940721043480553119510044341, −7.53313432835816202376978315834, −7.15506779281197965351716475533, −7.03494875052797441026295742556, −6.53528954375084791442464160975, −6.40850087658375716092715946670, −5.34677270563315951793799564245, −5.29409387424004825957703162325, −4.94179386008269118469647978069, −4.46032078340841994428082786354, −4.45926733662171269660191710541, −4.09954223669269929260750447000, −3.72280023376165040809829371667, −3.70304233461520988126906660880, −2.99693726393766606120851860447, −2.42326741041247151086863629066, −1.59394947702005839200279605791, −1.15100878527600699774350136015, 1.15100878527600699774350136015, 1.59394947702005839200279605791, 2.42326741041247151086863629066, 2.99693726393766606120851860447, 3.70304233461520988126906660880, 3.72280023376165040809829371667, 4.09954223669269929260750447000, 4.45926733662171269660191710541, 4.46032078340841994428082786354, 4.94179386008269118469647978069, 5.29409387424004825957703162325, 5.34677270563315951793799564245, 6.40850087658375716092715946670, 6.53528954375084791442464160975, 7.03494875052797441026295742556, 7.15506779281197965351716475533, 7.53313432835816202376978315834, 7.81940721043480553119510044341, 7.956946056206819531068158383000, 8.047008761313330853147939240620, 8.490467274549382096495902889743, 8.972750987382410267705541612372, 9.324141254380461169760587001755, 9.467994169589191855473599328590, 10.26982506078357238541825753478

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