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Properties

Label 4-27040-1.1-c1e2-0-0
Degree $4$
Conductor $27040$
Sign $1$
Analytic cond. $1.72409$
Root an. cond. $1.14588$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s + 5·13-s + 16-s − 3·17-s + 18-s − 20-s − 4·25-s + 5·26-s + 15·29-s + 32-s − 3·34-s + 36-s + 4·37-s − 40-s − 12·41-s − 45-s + 2·49-s − 4·50-s + 5·52-s − 12·53-s + 15·58-s + 7·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.38·13-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.223·20-s − 4/5·25-s + 0.980·26-s + 2.78·29-s + 0.176·32-s − 0.514·34-s + 1/6·36-s + 0.657·37-s − 0.158·40-s − 1.87·41-s − 0.149·45-s + 2/7·49-s − 0.565·50-s + 0.693·52-s − 1.64·53-s + 1.96·58-s + 0.896·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27040 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27040 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27040\)    =    \(2^{5} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1.72409\)
Root analytic conductor: \(1.14588\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27040,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.829143548\)
\(L(\frac12)\) \(\approx\) \(1.829143548\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.23.a_q
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.29.ap_ei
31$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.31.a_ar
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.ae_bq
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.m_de
43$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.43.a_ack
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.m_fd
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.ah_ek
67$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \) 2.67.a_act
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.l_gs
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.83.a_acn
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−10.73780202567534686219785697389, −10.12894942841913704777173096278, −9.757902806474674572231314430943, −8.808928364936345822631434152737, −8.406742742842020540802593447647, −8.021655451370116440669718003127, −7.18332951878531314974698003679, −6.59529237622968689411140696964, −6.26412828957952238561095946195, −5.50011109571356326963634847361, −4.65537232685169554222116895887, −4.24381142133795531659232255818, −3.48168648482626145852959327520, −2.72485959907788010563510238528, −1.45669789811999648128247048687, 1.45669789811999648128247048687, 2.72485959907788010563510238528, 3.48168648482626145852959327520, 4.24381142133795531659232255818, 4.65537232685169554222116895887, 5.50011109571356326963634847361, 6.26412828957952238561095946195, 6.59529237622968689411140696964, 7.18332951878531314974698003679, 8.021655451370116440669718003127, 8.406742742842020540802593447647, 8.808928364936345822631434152737, 9.757902806474674572231314430943, 10.12894942841913704777173096278, 10.73780202567534686219785697389

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