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Properties

Label 4-3408e2-1.1-c1e2-0-0
Degree $4$
Conductor $11614464$
Sign $1$
Analytic cond. $740.548$
Root an. cond. $5.21660$
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  − 2·3-s − 5·5-s + 4·7-s + 3·9-s + 8·11-s − 13-s + 10·15-s − 4·17-s + 8·19-s − 8·21-s + 3·23-s + 10·25-s − 4·27-s + 3·29-s − 8·31-s − 16·33-s − 20·35-s − 37-s + 2·39-s − 17·41-s + 15·43-s − 15·45-s − 15·47-s + 3·49-s + 8·51-s + 9·53-s − 40·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.23·5-s + 1.51·7-s + 9-s + 2.41·11-s − 0.277·13-s + 2.58·15-s − 0.970·17-s + 1.83·19-s − 1.74·21-s + 0.625·23-s + 2·25-s − 0.769·27-s + 0.557·29-s − 1.43·31-s − 2.78·33-s − 3.38·35-s − 0.164·37-s + 0.320·39-s − 2.65·41-s + 2.28·43-s − 2.23·45-s − 2.18·47-s + 3/7·49-s + 1.12·51-s + 1.23·53-s − 5.39·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11614464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11614464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11614464\)    =    \(2^{8} \cdot 3^{2} \cdot 71^{2}\)
Sign: $1$
Analytic conductor: \(740.548\)
Root analytic conductor: \(5.21660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11614464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.547784894\)
\(L(\frac12)\) \(\approx\) \(1.547784894\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
71$C_1$ \( ( 1 + T )^{2} \)
good5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.5.f_p
7$D_{4}$ \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_n
11$D_{4}$ \( 1 - 8 T + 3 p T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.11.ai_bh
13$D_{4}$ \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_p
17$D_{4}$ \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_bh
19$D_{4}$ \( 1 - 8 T + 49 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.19.ai_bx
23$D_{4}$ \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_bl
29$C_4$ \( 1 - 3 T - T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_ab
31$D_{4}$ \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cg
37$D_{4}$ \( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} \) 2.37.b_br
41$D_{4}$ \( 1 + 17 T + 153 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.41.r_fx
43$D_{4}$ \( 1 - 15 T + 131 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.43.ap_fb
47$D_{4}$ \( 1 + 15 T + 139 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.47.p_fj
53$D_{4}$ \( 1 - 9 T + 125 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_ev
59$D_{4}$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_ad
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.61.ak_fr
67$D_{4}$ \( 1 - 5 T - 11 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.67.af_al
73$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_fm
79$D_{4}$ \( 1 - 5 T + 163 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.79.af_gh
83$D_{4}$ \( 1 - 12 T + 197 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_hp
89$D_{4}$ \( 1 + 14 T + 207 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_hz
97$D_{4}$ \( 1 - 5 T + 139 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.97.af_fj
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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

−8.566895574336686341532281628374, −8.530635705674788266335818021149, −7.84833262692399986110274740309, −7.72861748543477130248396113957, −7.16880883513211192196862309299, −7.08664310226954563042216446217, −6.60683767318551020953830878095, −6.40861294810726719433466718549, −5.50472107811706613238129997922, −5.43721269986009702158609803237, −4.75509544335424691449037189980, −4.67105292440982797478732204269, −4.17480375470926985408209685667, −3.86439692611807571222961528028, −3.45181325800960392245730108069, −3.13089368022580925179178048034, −1.84869590124413878024897715859, −1.73715063629978224967945621350, −0.912571750520417726654302185450, −0.55438868112504039347535894667, 0.55438868112504039347535894667, 0.912571750520417726654302185450, 1.73715063629978224967945621350, 1.84869590124413878024897715859, 3.13089368022580925179178048034, 3.45181325800960392245730108069, 3.86439692611807571222961528028, 4.17480375470926985408209685667, 4.67105292440982797478732204269, 4.75509544335424691449037189980, 5.43721269986009702158609803237, 5.50472107811706613238129997922, 6.40861294810726719433466718549, 6.60683767318551020953830878095, 7.08664310226954563042216446217, 7.16880883513211192196862309299, 7.72861748543477130248396113957, 7.84833262692399986110274740309, 8.530635705674788266335818021149, 8.566895574336686341532281628374

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