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Properties

Label 4-7104e2-1.1-c1e2-0-4
Degree $4$
Conductor $50466816$
Sign $1$
Analytic cond. $3217.80$
Root an. cond. $7.53164$
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 − 2·5-s + 8·7-s + 3·9-s − 4·11-s − 4·15-s − 6·17-s + 16·21-s + 2·23-s − 2·25-s + 4·27-s + 6·29-s + 12·31-s − 8·33-s − 16·35-s − 2·37-s − 4·41-s − 4·43-s − 6·45-s + 16·47-s + 34·49-s − 12·51-s − 8·53-s + 8·55-s − 2·59-s + 12·61-s + 24·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 3.02·7-s + 9-s − 1.20·11-s − 1.03·15-s − 1.45·17-s + 3.49·21-s + 0.417·23-s − 2/5·25-s + 0.769·27-s + 1.11·29-s + 2.15·31-s − 1.39·33-s − 2.70·35-s − 0.328·37-s − 0.624·41-s − 0.609·43-s − 0.894·45-s + 2.33·47-s + 34/7·49-s − 1.68·51-s − 1.09·53-s + 1.07·55-s − 0.260·59-s + 1.53·61-s + 3.02·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50466816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50466816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50466816\)    =    \(2^{12} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(3217.80\)
Root analytic conductor: \(7.53164\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50466816,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.074205315\)
\(L(\frac12)\) \(\approx\) \(7.074205315\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
37$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_g
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_g
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bm
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_bq
29$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_w
31$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.31.am_da
41$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_g
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_dy
59$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_ag
61$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_da
67$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_cs
71$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_da
73$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_fm
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.79.a_da
83$D_{4}$ \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.83.au_jm
89$D_{4}$ \( 1 - 22 T + 254 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.89.aw_ju
97$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.97.a_o
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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

−8.007848097394803836245804388736, −7.967056724937739633346785864732, −7.63352810530751171340528994114, −7.30094615637211495946179395433, −6.77260131803474404347538822429, −6.66967289013105130252905046527, −5.81036123181801106775535576153, −5.68605909160995947560425241022, −4.80850042928208954514357297398, −4.79707847377956575041530449579, −4.60912046894601727277770151792, −4.47564009260210454382652147807, −3.60476445675497426722974691438, −3.53579542541740551306990580862, −2.82483123154982283510174397919, −2.31256287172358651828419250955, −2.10445211157148549757208776286, −1.85063700399191465051577669233, −0.957861644131929004822478972093, −0.70413817059782618212835187264, 0.70413817059782618212835187264, 0.957861644131929004822478972093, 1.85063700399191465051577669233, 2.10445211157148549757208776286, 2.31256287172358651828419250955, 2.82483123154982283510174397919, 3.53579542541740551306990580862, 3.60476445675497426722974691438, 4.47564009260210454382652147807, 4.60912046894601727277770151792, 4.79707847377956575041530449579, 4.80850042928208954514357297398, 5.68605909160995947560425241022, 5.81036123181801106775535576153, 6.66967289013105130252905046527, 6.77260131803474404347538822429, 7.30094615637211495946179395433, 7.63352810530751171340528994114, 7.967056724937739633346785864732, 8.007848097394803836245804388736

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