Properties

Label 4-2166e2-1.1-c3e2-0-7
Degree $4$
Conductor $4691556$
Sign $1$
Analytic cond. $16332.3$
Root an. cond. $11.3047$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 6·3-s + 12·4-s − 13·5-s + 24·6-s − 17·7-s + 32·8-s + 27·9-s − 52·10-s + 14·11-s + 72·12-s + 19·13-s − 68·14-s − 78·15-s + 80·16-s − 12·17-s + 108·18-s − 156·20-s − 102·21-s + 56·22-s − 54·23-s + 192·24-s − 92·25-s + 76·26-s + 108·27-s − 204·28-s − 130·29-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.16·5-s + 1.63·6-s − 0.917·7-s + 1.41·8-s + 9-s − 1.64·10-s + 0.383·11-s + 1.73·12-s + 0.405·13-s − 1.29·14-s − 1.34·15-s + 5/4·16-s − 0.171·17-s + 1.41·18-s − 1.74·20-s − 1.05·21-s + 0.542·22-s − 0.489·23-s + 1.63·24-s − 0.735·25-s + 0.573·26-s + 0.769·27-s − 1.37·28-s − 0.832·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4691556\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(16332.3\)
Root analytic conductor: \(11.3047\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2166} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4691556,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
3$C_1$ \( ( 1 - p T )^{2} \)
19 \( 1 \)
good5$D_{4}$ \( 1 + 13 T + 261 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 17 T + 697 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 14 T + 2631 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 19 T + 4453 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 12 T + 8417 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 54 T + 24743 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 130 T + 43758 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 239 T + 41461 T^{2} + 239 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 4 p T + 74782 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 331 T + 158201 T^{2} - 331 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 224 T + 160513 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 333 T + 234917 T^{2} - 333 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 766 T + 306663 T^{2} + 766 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 460 T + 238938 T^{2} + 460 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 494 T + 436846 T^{2} - 494 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 133 T + 585147 T^{2} + 133 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 459 T + 693461 T^{2} + 459 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 396 T + 810393 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 1000 T + 1157953 T^{2} - 1000 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 74 T + 1088763 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 1180 T + 1585058 T^{2} + 1180 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1938 T + 2745702 T^{2} + 1938 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.263422755456844582697810102977, −8.073453946281869208373139998544, −7.66205325662059052464281394390, −7.45137071620747141844427692051, −6.77492338296616773864077399924, −6.75575822488754397896106522873, −6.02506408762606047336468711322, −5.90852378574887775809878217043, −5.17303782526881701013523669880, −4.84176253361136468980567202819, −4.13694037839207776881827290322, −3.92231868785952042072083790861, −3.54196768744657859336427383116, −3.52262976430095927985305369391, −2.59916512531750744566313391627, −2.55245501851747760710249486406, −1.54258784359168848706279111674, −1.45363021171602645785431643037, 0, 0, 1.45363021171602645785431643037, 1.54258784359168848706279111674, 2.55245501851747760710249486406, 2.59916512531750744566313391627, 3.52262976430095927985305369391, 3.54196768744657859336427383116, 3.92231868785952042072083790861, 4.13694037839207776881827290322, 4.84176253361136468980567202819, 5.17303782526881701013523669880, 5.90852378574887775809878217043, 6.02506408762606047336468711322, 6.75575822488754397896106522873, 6.77492338296616773864077399924, 7.45137071620747141844427692051, 7.66205325662059052464281394390, 8.073453946281869208373139998544, 8.263422755456844582697810102977

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