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Properties

Label 4-1170e2-1.1-c1e2-0-11
Degree $4$
Conductor $1368900$
Sign $1$
Analytic cond. $87.2822$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 5-s − 3·6-s + 3·7-s − 8-s + 6·9-s + 10-s − 2·11-s − 13-s + 3·14-s − 3·15-s − 16-s + 6·18-s + 8·19-s − 9·21-s − 2·22-s − 7·23-s + 3·24-s − 26-s − 9·27-s − 3·29-s − 3·30-s − 4·31-s + 6·33-s + 3·35-s + 16·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 0.447·5-s − 1.22·6-s + 1.13·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s − 0.277·13-s + 0.801·14-s − 0.774·15-s − 1/4·16-s + 1.41·18-s + 1.83·19-s − 1.96·21-s − 0.426·22-s − 1.45·23-s + 0.612·24-s − 0.196·26-s − 1.73·27-s − 0.557·29-s − 0.547·30-s − 0.718·31-s + 1.04·33-s + 0.507·35-s + 2.63·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1368900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(87.2822\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1368900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.745188320\)
\(L(\frac12)\) \(\approx\) \(1.745188320\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_c
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.23.h_ba
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2^2$ \( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_acd
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.61.af_abk
67$C_2^2$ \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.67.j_o
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_v
83$C_2^2$ \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.83.ad_acw
89$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.89.ak_hv
97$C_2^2$ \( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.97.as_it
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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

−9.968055749591625790339597980684, −9.863697431340655582953628211976, −9.112431734958298960536335405609, −9.073167588980558745236048433163, −8.030943893458111425886585045022, −7.72331749214998438103396605118, −7.62626247318752613647235830766, −6.98831654149641468477673457876, −6.43188903676993378414737357580, −5.96172219759127693902626009365, −5.67108352846755912626316077745, −5.27264447022883889304099790371, −5.11350623427692859202350463922, −4.34194406748426048417201167592, −4.27696183524201837568361527369, −3.52700622958659314867488307578, −2.71599938367121663047557830645, −2.08546817410853292653440791962, −1.37675786589367605619153435946, −0.61274745139424892740520636734, 0.61274745139424892740520636734, 1.37675786589367605619153435946, 2.08546817410853292653440791962, 2.71599938367121663047557830645, 3.52700622958659314867488307578, 4.27696183524201837568361527369, 4.34194406748426048417201167592, 5.11350623427692859202350463922, 5.27264447022883889304099790371, 5.67108352846755912626316077745, 5.96172219759127693902626009365, 6.43188903676993378414737357580, 6.98831654149641468477673457876, 7.62626247318752613647235830766, 7.72331749214998438103396605118, 8.030943893458111425886585045022, 9.073167588980558745236048433163, 9.112431734958298960536335405609, 9.863697431340655582953628211976, 9.968055749591625790339597980684

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