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Properties

Label 4-117e2-1.1-c1e2-0-15
Degree $4$
Conductor $13689$
Sign $-1$
Analytic cond. $0.872822$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 2·9-s + 2·12-s − 5·13-s − 3·23-s + 25-s + 5·27-s − 12·29-s + 4·36-s + 5·39-s − 11·43-s + 5·49-s + 10·52-s + 12·53-s − 13·61-s + 8·64-s + 3·69-s − 75-s − 7·79-s + 81-s + 12·87-s + 6·92-s − 2·100-s + 27·101-s − 20·103-s + 12·107-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 2/3·9-s + 0.577·12-s − 1.38·13-s − 0.625·23-s + 1/5·25-s + 0.962·27-s − 2.22·29-s + 2/3·36-s + 0.800·39-s − 1.67·43-s + 5/7·49-s + 1.38·52-s + 1.64·53-s − 1.66·61-s + 64-s + 0.361·69-s − 0.115·75-s − 0.787·79-s + 1/9·81-s + 1.28·87-s + 0.625·92-s − 1/5·100-s + 2.68·101-s − 1.97·103-s + 1.16·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(0.872822\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 13689,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_2$ \( 1 + T + p T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.d_bu
29$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.m_dh
31$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.31.a_cd
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.37.a_bi
41$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.41.a_abl
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.43.l_ek
47$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.47.a_acd
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.53.am_fd
59$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \) 2.59.a_cb
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.61.n_fo
67$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \) 2.67.a_adl
71$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.71.a_du
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.73.a_bi
79$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.h_fu
83$C_2^2$ \( 1 + 125 T^{2} + p^{2} T^{4} \) 2.83.a_ev
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2^2$ \( 1 + 157 T^{2} + p^{2} T^{4} \) 2.97.a_gb
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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

−11.00532084754246167124084101708, −10.31453851596519414293283116870, −9.865541943633370090286786133496, −9.259193378129729411314699302122, −8.838535630268282341179855629225, −8.222861338133290193246821662946, −7.47671691054370486042388375079, −6.99355924731825873030171106118, −6.06697710305086175522084812220, −5.48114874742225891130904665830, −4.95287287063942728295349208421, −4.29189245265897939806533711216, −3.38434697765581707429375510649, −2.21350441470184917104183116839, 0, 2.21350441470184917104183116839, 3.38434697765581707429375510649, 4.29189245265897939806533711216, 4.95287287063942728295349208421, 5.48114874742225891130904665830, 6.06697710305086175522084812220, 6.99355924731825873030171106118, 7.47671691054370486042388375079, 8.222861338133290193246821662946, 8.838535630268282341179855629225, 9.259193378129729411314699302122, 9.865541943633370090286786133496, 10.31453851596519414293283116870, 11.00532084754246167124084101708

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