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Properties

Label 4-504e2-1.1-c1e2-0-10
Degree $4$
Conductor $254016$
Sign $1$
Analytic cond. $16.1962$
Root an. cond. $2.00610$
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  − 5-s − 5·7-s + 5·11-s + 4·13-s + 6·17-s − 2·19-s − 6·23-s + 5·25-s + 6·29-s − 5·31-s + 5·35-s + 2·37-s − 16·41-s − 8·43-s + 4·47-s + 18·49-s + 9·53-s − 5·55-s − 3·59-s + 12·61-s − 4·65-s − 2·67-s + 16·71-s + 14·73-s − 25·77-s − 79-s − 34·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.88·7-s + 1.50·11-s + 1.10·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s + 25-s + 1.11·29-s − 0.898·31-s + 0.845·35-s + 0.328·37-s − 2.49·41-s − 1.21·43-s + 0.583·47-s + 18/7·49-s + 1.23·53-s − 0.674·55-s − 0.390·59-s + 1.53·61-s − 0.496·65-s − 0.244·67-s + 1.89·71-s + 1.63·73-s − 2.84·77-s − 0.112·79-s − 3.73·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(254016\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(16.1962\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 254016,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.358371760\)
\(L(\frac12)\) \(\approx\) \(1.358371760\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_o
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_ag
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_abh
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.41.q_fq
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_abf
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_bc
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2^2$ \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_df
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.73.ao_et
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) 2.79.b_ada
83$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \) 2.83.bi_rn
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_jb
97$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.97.ag_hv
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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

−11.03588423748998573518481150375, −10.72272262441247440820082923254, −10.06504481196471554017482556095, −9.795498277928988663560234940374, −9.592779530101334843839807688943, −8.812730461102558982882346671003, −8.386693243709894395002389890294, −8.357897454813457825740921238065, −7.33901180689111044745534199649, −6.86946924839618668978276118211, −6.60937876214017638836146963732, −6.20259217815736382622553604487, −5.65879996575559199879377903551, −5.10631714560288618052034387382, −4.08445819281867512974779820046, −3.77954060820177748723425598295, −3.44287738734565821501010357804, −2.82758947239247120621671795459, −1.69489218550883750994371345228, −0.73584844925906169318956550176, 0.73584844925906169318956550176, 1.69489218550883750994371345228, 2.82758947239247120621671795459, 3.44287738734565821501010357804, 3.77954060820177748723425598295, 4.08445819281867512974779820046, 5.10631714560288618052034387382, 5.65879996575559199879377903551, 6.20259217815736382622553604487, 6.60937876214017638836146963732, 6.86946924839618668978276118211, 7.33901180689111044745534199649, 8.357897454813457825740921238065, 8.386693243709894395002389890294, 8.812730461102558982882346671003, 9.592779530101334843839807688943, 9.795498277928988663560234940374, 10.06504481196471554017482556095, 10.72272262441247440820082923254, 11.03588423748998573518481150375

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