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Properties

Label 4-2808e2-1.1-c1e2-0-19
Degree $4$
Conductor $7884864$
Sign $1$
Analytic cond. $502.745$
Root an. cond. $4.73518$
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  − 3·5-s + 5·7-s − 7·11-s + 2·13-s − 8·17-s + 2·19-s + 23-s + 25-s + 7·29-s − 31-s − 15·35-s + 37-s − 8·41-s − 12·43-s − 5·47-s + 9·49-s − 53-s + 21·55-s + 4·59-s − 25·61-s − 6·65-s + 4·67-s − 3·71-s − 24·73-s − 35·77-s + 16·79-s − 18·83-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.88·7-s − 2.11·11-s + 0.554·13-s − 1.94·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s + 1.29·29-s − 0.179·31-s − 2.53·35-s + 0.164·37-s − 1.24·41-s − 1.82·43-s − 0.729·47-s + 9/7·49-s − 0.137·53-s + 2.83·55-s + 0.520·59-s − 3.20·61-s − 0.744·65-s + 0.488·67-s − 0.356·71-s − 2.80·73-s − 3.98·77-s + 1.80·79-s − 1.97·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7884864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7884864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7884864\)    =    \(2^{6} \cdot 3^{6} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(502.745\)
Root analytic conductor: \(4.73518\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7884864,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
7$D_{4}$ \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.7.af_q
11$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.11.h_be
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_w
23$D_{4}$ \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_i
29$D_{4}$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.29.ah_bg
31$D_{4}$ \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_y
37$D_{4}$ \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) 2.37.ab_bk
41$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_be
43$D_{4}$ \( 1 + 12 T + 105 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_eb
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_ck
53$D_{4}$ \( 1 + T + 102 T^{2} + p T^{3} + p^{2} T^{4} \) 2.53.b_dy
59$D_{4}$ \( 1 - 4 T + 105 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_eb
61$D_{4}$ \( 1 + 25 T + 274 T^{2} + 25 p T^{3} + p^{2} T^{4} \) 2.61.z_ko
67$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_cs
71$D_{4}$ \( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.71.d_ec
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.73.y_le
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$D_{4}$ \( 1 + 18 T + 179 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_gx
89$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.89.c_gx
97$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.97.a_ada
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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

−8.398832495251478827232732214135, −8.290991047036757997709273293975, −7.961906441728662115920940131962, −7.61535106580998324094938792885, −7.07624949049831260516644937760, −7.06433575423158418735995593496, −6.23247010404546045564286850038, −5.98492608229808507628673856736, −5.29379124278750908746513348198, −4.87669998411317802149597655784, −4.67902431801573317527577020400, −4.59379469499948889398207946036, −3.80861742399559882211207823474, −3.44689735577252670140106788892, −2.64506894971593902596127459427, −2.61859105865621800042631647147, −1.59496940392044983676594417019, −1.46351741509729371177242022736, 0, 0, 1.46351741509729371177242022736, 1.59496940392044983676594417019, 2.61859105865621800042631647147, 2.64506894971593902596127459427, 3.44689735577252670140106788892, 3.80861742399559882211207823474, 4.59379469499948889398207946036, 4.67902431801573317527577020400, 4.87669998411317802149597655784, 5.29379124278750908746513348198, 5.98492608229808507628673856736, 6.23247010404546045564286850038, 7.06433575423158418735995593496, 7.07624949049831260516644937760, 7.61535106580998324094938792885, 7.961906441728662115920940131962, 8.290991047036757997709273293975, 8.398832495251478827232732214135

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