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Properties

Label 4-52e4-1.1-c1e2-0-7
Degree $4$
Conductor $7311616$
Sign $1$
Analytic cond. $466.194$
Root an. cond. $4.64667$
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  + 3-s + 5-s + 5·7-s − 9-s + 2·11-s + 15-s + 3·17-s − 4·19-s + 5·21-s + 2·23-s − 5·25-s − 2·29-s + 2·31-s + 2·33-s + 5·35-s − 15·37-s − 16·41-s + 15·43-s − 45-s − 11·47-s + 9·49-s + 3·51-s + 8·53-s + 2·55-s − 4·57-s + 20·59-s + 14·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.88·7-s − 1/3·9-s + 0.603·11-s + 0.258·15-s + 0.727·17-s − 0.917·19-s + 1.09·21-s + 0.417·23-s − 25-s − 0.371·29-s + 0.359·31-s + 0.348·33-s + 0.845·35-s − 2.46·37-s − 2.49·41-s + 2.28·43-s − 0.149·45-s − 1.60·47-s + 9/7·49-s + 0.420·51-s + 1.09·53-s + 0.269·55-s − 0.529·57-s + 2.60·59-s + 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7311616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7311616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7311616\)    =    \(2^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(466.194\)
Root analytic conductor: \(4.64667\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7311616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.499799204\)
\(L(\frac12)\) \(\approx\) \(4.499799204\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_g
7$D_{4}$ \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.7.af_q
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_g
17$D_{4}$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_bg
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_be
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_bq
31$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_bu
37$D_{4}$ \( 1 + 15 T + 126 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.37.p_ew
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.41.q_fq
43$D_{4}$ \( 1 - 15 T + 138 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.43.ap_fi
47$D_{4}$ \( 1 + 11 T + 120 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.47.l_eq
53$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_cc
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.59.au_ik
61$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.61.ao_fy
67$D_{4}$ \( 1 - 10 T + 142 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_fm
71$D_{4}$ \( 1 + T + 104 T^{2} + p T^{3} + p^{2} T^{4} \) 2.71.b_ea
73$D_{4}$ \( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_c
79$D_{4}$ \( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.79.ao_hi
83$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.83.a_du
89$D_{4}$ \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_gg
97$D_{4}$ \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_fa
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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

−8.696771018842542207263351075491, −8.628896311861460463039910894848, −8.312877077933587342902511661992, −8.160674307231356457011931974376, −7.55623501111772194368621232307, −7.17257939692592664729694541977, −6.76170208878051368246338961100, −6.50716171125282464990685070156, −5.78926185900397933972520926536, −5.45787228100373829980409662941, −5.03119289509308449322608657382, −5.00465887704030808227982046069, −4.01441604485184811788092879351, −4.00027731792265870510206060330, −3.44526309235531808256193394657, −2.82555382675307408894006694176, −2.15359402162901983017170993996, −1.90127515572272056260445980288, −1.48149610303832195097425302985, −0.66267037626513989326510550280, 0.66267037626513989326510550280, 1.48149610303832195097425302985, 1.90127515572272056260445980288, 2.15359402162901983017170993996, 2.82555382675307408894006694176, 3.44526309235531808256193394657, 4.00027731792265870510206060330, 4.01441604485184811788092879351, 5.00465887704030808227982046069, 5.03119289509308449322608657382, 5.45787228100373829980409662941, 5.78926185900397933972520926536, 6.50716171125282464990685070156, 6.76170208878051368246338961100, 7.17257939692592664729694541977, 7.55623501111772194368621232307, 8.160674307231356457011931974376, 8.312877077933587342902511661992, 8.628896311861460463039910894848, 8.696771018842542207263351075491

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