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Properties

Label 4-608e2-1.1-c1e2-0-4
Degree $4$
Conductor $369664$
Sign $1$
Analytic cond. $23.5700$
Root an. cond. $2.20338$
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-s − 5-s + 6·7-s − 9-s − 3·11-s + 13-s − 15-s − 8·17-s + 2·19-s + 6·21-s + 7·23-s − 5·25-s − 3·29-s + 10·31-s − 3·33-s − 6·35-s + 6·37-s + 39-s + 8·41-s + 15·43-s + 45-s + 5·47-s + 13·49-s − 8·51-s − 13·53-s + 3·55-s + 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 2.26·7-s − 1/3·9-s − 0.904·11-s + 0.277·13-s − 0.258·15-s − 1.94·17-s + 0.458·19-s + 1.30·21-s + 1.45·23-s − 25-s − 0.557·29-s + 1.79·31-s − 0.522·33-s − 1.01·35-s + 0.986·37-s + 0.160·39-s + 1.24·41-s + 2.28·43-s + 0.149·45-s + 0.729·47-s + 13/7·49-s − 1.12·51-s − 1.78·53-s + 0.404·55-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(369664\)    =    \(2^{10} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(23.5700\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 369664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506155261\)
\(L(\frac12)\) \(\approx\) \(2.506155261\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
5$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_g
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.7.ag_x
11$D_{4}$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_u
13$D_{4}$ \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_w
17$D_{4}$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bh
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.23.ah_cc
29$D_{4}$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_w
31$D_{4}$ \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_cs
37$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_co
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.41.ai_du
43$D_{4}$ \( 1 - 15 T + 138 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.43.ap_fi
47$D_{4}$ \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.47.af_ck
53$D_{4}$ \( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.53.n_fo
59$D_{4}$ \( 1 + 13 T + 156 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.59.n_ga
61$D_{4}$ \( 1 - 9 T + 36 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.61.aj_bk
67$D_{4}$ \( 1 + 5 T + 136 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.67.f_fg
71$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_dm
73$D_{4}$ \( 1 + 2 T + 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_db
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$D_{4}$ \( 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_bm
89$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_bi
97$D_{4}$ \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_he
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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

−11.05653179784718687030000414234, −10.76049969597573691655238784929, −9.967591247771858143259641828511, −9.392410201670798868356218425549, −8.980582610518813772532807858459, −8.665732770087886597582778248719, −8.034577044975961632511233168401, −7.992761883473664672575546172731, −7.49629067270974422852508811169, −7.14734230998654526660491502729, −6.11417336287529391942224171467, −6.06238245304157854839980263678, −4.96591759499945842784207650948, −4.93087888129573414247699269601, −4.38320622430800111044171559562, −3.91433566893515775066259090814, −2.86011579749282015240533827551, −2.53346238926307565915139486861, −1.85961794065305034181282345647, −0.896188645083860498684360941652, 0.896188645083860498684360941652, 1.85961794065305034181282345647, 2.53346238926307565915139486861, 2.86011579749282015240533827551, 3.91433566893515775066259090814, 4.38320622430800111044171559562, 4.93087888129573414247699269601, 4.96591759499945842784207650948, 6.06238245304157854839980263678, 6.11417336287529391942224171467, 7.14734230998654526660491502729, 7.49629067270974422852508811169, 7.992761883473664672575546172731, 8.034577044975961632511233168401, 8.665732770087886597582778248719, 8.980582610518813772532807858459, 9.392410201670798868356218425549, 9.967591247771858143259641828511, 10.76049969597573691655238784929, 11.05653179784718687030000414234

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