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Properties

Label 4-3654e2-1.1-c1e2-0-2
Degree $4$
Conductor $13351716$
Sign $1$
Analytic cond. $851.316$
Root an. cond. $5.40160$
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  − 4-s − 4·5-s − 2·7-s + 4·13-s + 16-s + 4·20-s + 12·23-s + 2·25-s + 2·28-s + 4·29-s + 8·35-s + 3·49-s − 4·52-s + 16·53-s − 16·59-s − 64-s − 16·65-s − 8·67-s − 4·71-s − 4·80-s − 32·83-s − 8·91-s − 12·92-s − 2·100-s + 8·103-s − 28·107-s − 36·109-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s − 0.755·7-s + 1.10·13-s + 1/4·16-s + 0.894·20-s + 2.50·23-s + 2/5·25-s + 0.377·28-s + 0.742·29-s + 1.35·35-s + 3/7·49-s − 0.554·52-s + 2.19·53-s − 2.08·59-s − 1/8·64-s − 1.98·65-s − 0.977·67-s − 0.474·71-s − 0.447·80-s − 3.51·83-s − 0.838·91-s − 1.25·92-s − 1/5·100-s + 0.788·103-s − 2.70·107-s − 3.44·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13351716 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13351716 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13351716\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(851.316\)
Root analytic conductor: \(5.40160\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13351716,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3505357432\)
\(L(\frac12)\) \(\approx\) \(0.3505357432\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
29$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.41.a_ada
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.53.aq_go
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.61.a_aeo
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.71.e_fq
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.83.bg_qg
89$C_2^2$ \( 1 - 174 T^{2} + p^{2} T^{4} \) 2.89.a_ags
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−8.898039231181463492435128849424, −8.496618474873186071655816998977, −8.022456425533532783047085129639, −7.48287679165733303619613175061, −7.41056376594535506819627682266, −6.93481132231521312814331749609, −6.63889830567276750009890291212, −6.04907551410970446755937417953, −5.83893154754430655859080133759, −5.21197072508469921628679374344, −4.91552094777684829492894669011, −4.35937182478706404543176800831, −4.03225237605018161961234303240, −3.76518590747653290770576715337, −3.35842512617870981390284114698, −2.74628649096403261060761440718, −2.69926644537589361240754875041, −1.28340812781489570559167210247, −1.22154529781302261322104678242, −0.20684018429380617775513423995, 0.20684018429380617775513423995, 1.22154529781302261322104678242, 1.28340812781489570559167210247, 2.69926644537589361240754875041, 2.74628649096403261060761440718, 3.35842512617870981390284114698, 3.76518590747653290770576715337, 4.03225237605018161961234303240, 4.35937182478706404543176800831, 4.91552094777684829492894669011, 5.21197072508469921628679374344, 5.83893154754430655859080133759, 6.04907551410970446755937417953, 6.63889830567276750009890291212, 6.93481132231521312814331749609, 7.41056376594535506819627682266, 7.48287679165733303619613175061, 8.022456425533532783047085129639, 8.496618474873186071655816998977, 8.898039231181463492435128849424

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