Properties

Label 4-21e4-1.1-c1e2-0-17
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $12.4002$
Root an. cond. $1.87654$
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  − 3·2-s + 3·3-s + 4·4-s + 6·5-s − 9·6-s − 3·8-s + 6·9-s − 18·10-s + 12·12-s + 3·13-s + 18·15-s + 3·16-s − 3·17-s − 18·18-s + 9·19-s + 24·20-s − 9·24-s + 17·25-s − 9·26-s + 9·27-s − 9·29-s − 54·30-s + 6·31-s − 6·32-s + 9·34-s + 24·36-s − 7·37-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.73·3-s + 2·4-s + 2.68·5-s − 3.67·6-s − 1.06·8-s + 2·9-s − 5.69·10-s + 3.46·12-s + 0.832·13-s + 4.64·15-s + 3/4·16-s − 0.727·17-s − 4.24·18-s + 2.06·19-s + 5.36·20-s − 1.83·24-s + 17/5·25-s − 1.76·26-s + 1.73·27-s − 1.67·29-s − 9.85·30-s + 1.07·31-s − 1.06·32-s + 1.54·34-s + 4·36-s − 1.15·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(12.4002\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.783580157\)
\(L(\frac12)\) \(\approx\) \(1.783580157\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 - p T + p T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 15 T + 128 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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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

−10.71430142050717045548279579408, −10.70704290299500632395616395743, −9.966812068373971656602248906909, −9.622970219408949043899417321996, −9.489066931635769291659851217351, −9.339725434933373327355123457551, −8.758203295864613541670595723170, −8.508188353993272298851908614266, −7.87622207116831882661162770878, −7.55199664669973416981709926685, −6.95026303310152955015798327225, −6.34762994543872814409830464126, −5.88170551328094744574311991043, −5.33625661556151440208566839580, −4.60296041835022587449067178750, −3.38975232692628683184736471929, −3.10915928561486700508227992890, −2.18008246770824568750489151264, −1.68506314583233985598526436150, −1.31476918920905115884516405007, 1.31476918920905115884516405007, 1.68506314583233985598526436150, 2.18008246770824568750489151264, 3.10915928561486700508227992890, 3.38975232692628683184736471929, 4.60296041835022587449067178750, 5.33625661556151440208566839580, 5.88170551328094744574311991043, 6.34762994543872814409830464126, 6.95026303310152955015798327225, 7.55199664669973416981709926685, 7.87622207116831882661162770878, 8.508188353993272298851908614266, 8.758203295864613541670595723170, 9.339725434933373327355123457551, 9.489066931635769291659851217351, 9.622970219408949043899417321996, 9.966812068373971656602248906909, 10.70704290299500632395616395743, 10.71430142050717045548279579408

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