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Properties

Label 4-9075e2-1.1-c1e2-0-30
Degree $4$
Conductor $82355625$
Sign $1$
Analytic cond. $5251.06$
Root an. cond. $8.51259$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 2·4-s − 2·6-s + 6·7-s − 3·8-s + 3·9-s + 4·12-s + 3·13-s + 6·14-s + 16-s + 6·17-s + 3·18-s + 10·19-s − 12·21-s − 13·23-s + 6·24-s + 3·26-s − 4·27-s − 12·28-s − 5·29-s − 11·31-s + 2·32-s + 6·34-s − 6·36-s − 11·37-s + 10·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 4-s − 0.816·6-s + 2.26·7-s − 1.06·8-s + 9-s + 1.15·12-s + 0.832·13-s + 1.60·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s + 2.29·19-s − 2.61·21-s − 2.71·23-s + 1.22·24-s + 0.588·26-s − 0.769·27-s − 2.26·28-s − 0.928·29-s − 1.97·31-s + 0.353·32-s + 1.02·34-s − 36-s − 1.80·37-s + 1.62·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(82355625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5251.06\)
Root analytic conductor: \(8.51259\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 82355625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
11 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.7.ag_x
13$D_{4}$ \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_r
17$D_{4}$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_x
19$D_{4}$ \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.19.ak_cg
23$D_{4}$ \( 1 + 13 T + 87 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.23.n_dj
29$D_{4}$ \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.29.f_cl
31$C_4$ \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.31.l_dd
37$D_{4}$ \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.37.l_dp
41$D_{4}$ \( 1 + 4 T + 81 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_dd
43$D_{4}$ \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_dt
47$D_{4}$ \( 1 + 21 T + 203 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.47.v_hv
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_dy
59$D_{4}$ \( 1 + 15 T + 163 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.59.p_gh
61$D_{4}$ \( 1 - 16 T + 181 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.61.aq_gz
67$D_{4}$ \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_dp
71$D_{4}$ \( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_fb
73$D_{4}$ \( 1 + 7 T + 147 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.73.h_fr
79$D_{4}$ \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.79.f_cl
83$D_{4}$ \( 1 - 23 T + 297 T^{2} - 23 p T^{3} + p^{2} T^{4} \) 2.83.ax_ll
89$D_{4}$ \( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.89.p_fd
97$D_{4}$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_x
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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

−7.61654525827892912681544626671, −7.42442649510917366155846275444, −6.76849024713045363299505953477, −6.43735114878838117891939347056, −5.81992966520887133200303347149, −5.67961599843621001923517075958, −5.36342494109812180066380740425, −5.23028755863539925821278741466, −4.87417666107340018015638331922, −4.63183694104525777612342143144, −4.02146284315080136480662303728, −3.80075802093693281974093813257, −3.49725412718601042659306848743, −3.20919893245594078758774762000, −2.23335032343851284555719073242, −1.65295468296667353948395928053, −1.40417916078730658830060001482, −1.27159492024378143510262470889, 0, 0, 1.27159492024378143510262470889, 1.40417916078730658830060001482, 1.65295468296667353948395928053, 2.23335032343851284555719073242, 3.20919893245594078758774762000, 3.49725412718601042659306848743, 3.80075802093693281974093813257, 4.02146284315080136480662303728, 4.63183694104525777612342143144, 4.87417666107340018015638331922, 5.23028755863539925821278741466, 5.36342494109812180066380740425, 5.67961599843621001923517075958, 5.81992966520887133200303347149, 6.43735114878838117891939347056, 6.76849024713045363299505953477, 7.42442649510917366155846275444, 7.61654525827892912681544626671

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