Properties

Label 4-1014e2-1.1-c1e2-0-4
Degree $4$
Conductor $1028196$
Sign $1$
Analytic cond. $65.5586$
Root an. cond. $2.84549$
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-s − 3-s − 6·5-s + 6-s + 2·7-s + 8-s + 6·10-s + 6·11-s − 2·14-s + 6·15-s − 16-s + 3·17-s + 2·19-s − 2·21-s − 6·22-s + 6·23-s − 24-s + 17·25-s + 27-s − 3·29-s − 6·30-s + 8·31-s − 6·33-s − 3·34-s − 12·35-s − 7·37-s − 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 2.68·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1.89·10-s + 1.80·11-s − 0.534·14-s + 1.54·15-s − 1/4·16-s + 0.727·17-s + 0.458·19-s − 0.436·21-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 17/5·25-s + 0.192·27-s − 0.557·29-s − 1.09·30-s + 1.43·31-s − 1.04·33-s − 0.514·34-s − 2.02·35-s − 1.15·37-s − 0.324·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1028196\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(65.5586\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1014} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1028196,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8204895162\)
\(L(\frac12)\) \(\approx\) \(0.8204895162\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
13 \( 1 \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−10.17098017482769254764825271993, −9.749759990523999846255781611579, −9.020612880436558334322838123225, −9.001977840954471147593601840267, −8.466335345396784134129871665618, −7.943477552843235372051692402466, −7.83575555622262577170252828989, −7.40517446390885702761783735686, −6.81881506682350185880462802372, −6.71231790152800578131661867995, −5.98771638121019862726517837465, −5.11876807590698223064728580114, −5.01211841824957368727230080658, −4.27192319431339598584312167493, −4.03299847662459628965932896770, −3.48852194824463752075217887845, −3.16747251500587658017099786223, −1.94859706041883630553691688647, −0.965010919160016085248344419349, −0.68828884815438676489826169522, 0.68828884815438676489826169522, 0.965010919160016085248344419349, 1.94859706041883630553691688647, 3.16747251500587658017099786223, 3.48852194824463752075217887845, 4.03299847662459628965932896770, 4.27192319431339598584312167493, 5.01211841824957368727230080658, 5.11876807590698223064728580114, 5.98771638121019862726517837465, 6.71231790152800578131661867995, 6.81881506682350185880462802372, 7.40517446390885702761783735686, 7.83575555622262577170252828989, 7.943477552843235372051692402466, 8.466335345396784134129871665618, 9.001977840954471147593601840267, 9.020612880436558334322838123225, 9.749759990523999846255781611579, 10.17098017482769254764825271993

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