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Properties

Label 4-95e4-1.1-c1e2-0-6
Degree $4$
Conductor $81450625$
Sign $1$
Analytic cond. $5193.36$
Root an. cond. $8.48910$
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  + 4·3-s + 4-s + 2·7-s + 6·9-s + 6·11-s + 4·12-s + 3·13-s − 3·16-s + 17-s + 8·21-s − 4·27-s + 2·28-s − 3·29-s + 12·31-s + 24·33-s + 6·36-s + 11·37-s + 12·39-s + 3·41-s − 8·43-s + 6·44-s − 14·47-s − 12·48-s − 6·49-s + 4·51-s + 3·52-s + 11·53-s + ⋯
L(s)  = 1  + 2.30·3-s + 1/2·4-s + 0.755·7-s + 2·9-s + 1.80·11-s + 1.15·12-s + 0.832·13-s − 3/4·16-s + 0.242·17-s + 1.74·21-s − 0.769·27-s + 0.377·28-s − 0.557·29-s + 2.15·31-s + 4.17·33-s + 36-s + 1.80·37-s + 1.92·39-s + 0.468·41-s − 1.21·43-s + 0.904·44-s − 2.04·47-s − 1.73·48-s − 6/7·49-s + 0.560·51-s + 0.416·52-s + 1.51·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81450625\)    =    \(5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(5193.36\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 81450625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(13.55364331\)
\(L(\frac12)\) \(\approx\) \(13.55364331\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5 \( 1 \)
19 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_k
11$C_4$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_ba
13$D_{4}$ \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_r
17$D_{4}$ \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_bh
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$D_{4}$ \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_bd
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$D_{4}$ \( 1 - 11 T + 93 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.37.al_dp
41$D_{4}$ \( 1 - 3 T + 53 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_cb
43$D_{4}$ \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_de
47$D_{4}$ \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.47.o_fi
53$D_{4}$ \( 1 - 11 T + 105 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.53.al_eb
59$D_{4}$ \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_ek
61$D_{4}$ \( 1 - 13 T + 153 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.61.an_fx
67$D_{4}$ \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_fa
71$D_{4}$ \( 1 - 2 T + 138 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_fi
73$D_{4}$ \( 1 - 9 T + 65 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.73.aj_cn
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.79.e_gg
83$D_{4}$ \( 1 - 6 T + 130 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_fa
89$D_{4}$ \( 1 + T + 117 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_en
97$D_{4}$ \( 1 + 11 T + 193 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.97.l_hl
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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

−8.006320665930130547081588906133, −7.84567053183210594336980460417, −7.27023676897772042536200799632, −6.89461646952033672066651631672, −6.71736474138147691284895078986, −6.17022431635392131465161080427, −6.11765142509953107341377678580, −5.57868449882991614651330191671, −4.86537552009320521786910250041, −4.77810777971759326843641108417, −4.12489666866118289696953721317, −3.93314589604097448878423552544, −3.60022980967259409330785924700, −3.21122708075321220728376152156, −2.85184632573946110049644129959, −2.39193696113166877698221588979, −2.11728694068608996809464319744, −1.65689531816417491680861591721, −1.27990616664932516970826645625, −0.68606397342114157715312823560, 0.68606397342114157715312823560, 1.27990616664932516970826645625, 1.65689531816417491680861591721, 2.11728694068608996809464319744, 2.39193696113166877698221588979, 2.85184632573946110049644129959, 3.21122708075321220728376152156, 3.60022980967259409330785924700, 3.93314589604097448878423552544, 4.12489666866118289696953721317, 4.77810777971759326843641108417, 4.86537552009320521786910250041, 5.57868449882991614651330191671, 6.11765142509953107341377678580, 6.17022431635392131465161080427, 6.71736474138147691284895078986, 6.89461646952033672066651631672, 7.27023676897772042536200799632, 7.84567053183210594336980460417, 8.006320665930130547081588906133

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