↑

Properties

Label 4-18e2-1.1-c14e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $500.829$
Root an. cond. $4.73066$
Motivic weight $14$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8.19e3·4-s − 5.22e5·7-s + 1.33e8·13-s + 6.71e7·16-s + 1.19e9·19-s + 1.20e10·25-s + 4.27e9·28-s + 7.87e10·31-s + 2.86e11·37-s − 2.76e11·43-s − 1.15e12·49-s − 1.09e12·52-s + 4.00e12·61-s − 5.49e11·64-s + 1.44e13·67-s + 2.29e13·73-s − 9.76e12·76-s + 4.12e12·79-s − 6.96e13·91-s − 2.77e14·97-s − 9.86e13·100-s − 3.04e14·103-s + 2.46e14·109-s − 3.50e13·112-s + 6.65e14·121-s − 6.45e14·124-s + 127-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.634·7-s + 2.12·13-s + 1/4·16-s + 1.33·19-s + 1.97·25-s + 0.317·28-s + 2.86·31-s + 3.02·37-s − 1.01·43-s − 1.69·49-s − 1.06·52-s + 1.27·61-s − 1/8·64-s + 2.37·67-s + 2.08·73-s − 0.666·76-s + 0.214·79-s − 1.34·91-s − 3.43·97-s − 0.986·100-s − 2.47·103-s + 1.34·109-s − 0.158·112-s + 1.75·121-s − 1.43·124-s − 0.845·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(500.829\)
Root analytic conductor: \(4.73066\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(3.294328260\)
\(L(\frac12)\) \(\approx\) \(3.294328260\)
\(L(8)\) 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$C_2$ \( 1 + p^{13} T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 19267712 p^{4} T^{2} + p^{28} T^{4} \)
7$C_2$ \( ( 1 + 261076 T + p^{14} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 5501425876994 p^{2} T^{2} + p^{28} T^{4} \)
13$C_2$ \( ( 1 - 394808 p^{2} T + p^{14} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 263211151473265376 T^{2} + p^{28} T^{4} \)
19$C_2$ \( ( 1 - 595848944 T + p^{14} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 1356409974309041470 T^{2} + p^{28} T^{4} \)
29$C_2^2$ \( 1 - \)\(23\!\cdots\!04\)\( T^{2} + p^{28} T^{4} \)
31$C_2$ \( ( 1 - 39382968284 T + p^{14} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 143474668814 T + p^{14} T^{2} )^{2} \)
41$C_2^2$ \( 1 - \)\(60\!\cdots\!44\)\( T^{2} + p^{28} T^{4} \)
43$C_2$ \( ( 1 + 138451793800 T + p^{14} T^{2} )^{2} \)
47$C_2^2$ \( 1 - \)\(42\!\cdots\!38\)\( T^{2} + p^{28} T^{4} \)
53$C_2^2$ \( 1 - \)\(11\!\cdots\!60\)\( T^{2} + p^{28} T^{4} \)
59$C_2^2$ \( 1 + \)\(93\!\cdots\!50\)\( T^{2} + p^{28} T^{4} \)
61$C_2$ \( ( 1 - 2000892201890 T + p^{14} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7203353671592 T + p^{14} T^{2} )^{2} \)
71$C_2^2$ \( 1 - \)\(84\!\cdots\!70\)\( T^{2} + p^{28} T^{4} \)
73$C_2$ \( ( 1 - 11493961843376 T + p^{14} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 2060521096964 T + p^{14} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(14\!\cdots\!86\)\( T^{2} + p^{28} T^{4} \)
89$C_2^2$ \( 1 - \)\(33\!\cdots\!40\)\( T^{2} + p^{28} T^{4} \)
97$C_2$ \( ( 1 + 138795118452976 T + p^{14} T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.65605580598855960828388598751, −15.02975838181071716844719765709, −14.10217624523869239065434539250, −13.59701714586204510031505994291, −13.07036746809512345697637713484, −12.43152340598269612946441758561, −11.35588327627214341976197637652, −11.03629101678118853287105511941, −9.809716166795877369383676876355, −9.586960317040058427225426937739, −8.287807713613237135691271622001, −8.265452001680645299522931663984, −6.69004101098562820451397640708, −6.31116403591649882467273046277, −5.25385345145741415189490913258, −4.34722212267013955051257604574, −3.40192478218787167205171619302, −2.75493679077545165222041813001, −0.991075469036671992046651450331, −0.915535630128945524756755113677, 0.915535630128945524756755113677, 0.991075469036671992046651450331, 2.75493679077545165222041813001, 3.40192478218787167205171619302, 4.34722212267013955051257604574, 5.25385345145741415189490913258, 6.31116403591649882467273046277, 6.69004101098562820451397640708, 8.265452001680645299522931663984, 8.287807713613237135691271622001, 9.586960317040058427225426937739, 9.809716166795877369383676876355, 11.03629101678118853287105511941, 11.35588327627214341976197637652, 12.43152340598269612946441758561, 13.07036746809512345697637713484, 13.59701714586204510031505994291, 14.10217624523869239065434539250, 15.02975838181071716844719765709, 15.65605580598855960828388598751

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