Properties

Label 4-290e2-1.1-c1e2-0-5
Degree $4$
Conductor $84100$
Sign $1$
Analytic cond. $5.36228$
Root an. cond. $1.52172$
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  − 2·2-s + 3-s + 3·4-s − 2·5-s − 2·6-s + 5·7-s − 4·8-s − 2·9-s + 4·10-s − 2·11-s + 3·12-s + 9·13-s − 10·14-s − 2·15-s + 5·16-s + 3·17-s + 4·18-s + 6·19-s − 6·20-s + 5·21-s + 4·22-s + 7·23-s − 4·24-s + 3·25-s − 18·26-s − 2·27-s + 15·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.894·5-s − 0.816·6-s + 1.88·7-s − 1.41·8-s − 2/3·9-s + 1.26·10-s − 0.603·11-s + 0.866·12-s + 2.49·13-s − 2.67·14-s − 0.516·15-s + 5/4·16-s + 0.727·17-s + 0.942·18-s + 1.37·19-s − 1.34·20-s + 1.09·21-s + 0.852·22-s + 1.45·23-s − 0.816·24-s + 3/5·25-s − 3.53·26-s − 0.384·27-s + 2.83·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84100\)    =    \(2^{2} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(5.36228\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 84100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.105326979\)
\(L(\frac12)\) \(\approx\) \(1.105326979\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_d
7$D_{4}$ \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.7.af_r
11$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_k
13$D_{4}$ \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.13.aj_br
17$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_h
19$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_bi
23$D_{4}$ \( 1 - 7 T + 55 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.23.ah_cd
31$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_bn
37$D_{4}$ \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abq
41$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_dq
43$D_{4}$ \( 1 - 3 T + 85 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.43.ad_dh
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$D_{4}$ \( 1 - 3 T + 79 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_db
59$D_{4}$ \( 1 - 9 T + 109 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_ef
61$D_{4}$ \( 1 + 17 T + 165 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.61.r_gj
67$D_{4}$ \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_di
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$D_{4}$ \( 1 + 13 T + 107 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.73.n_ed
79$D_{4}$ \( 1 - T + 129 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ez
83$D_{4}$ \( 1 + 10 T + 178 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_gw
89$D_{4}$ \( 1 + 22 T + 286 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.89.w_la
97$D_{4}$ \( 1 - 13 T + 207 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_hz
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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

−11.64448853237840978152738087403, −11.44715470203241819617330928940, −10.93880015048394908629159940051, −10.87642575347050855324411838616, −10.24020819636558820818875281638, −9.535443640693436142856681132436, −8.765082493453845884582129301720, −8.689362052218756979442645513614, −8.319202498085562194570074724910, −7.995271560601890934578164778827, −7.25526374973238832119327326179, −7.25043011485310189304278098122, −6.15149921353648408473937998277, −5.53926944552609927239784005951, −5.09767152865771135713722114004, −4.17540590010277769479068710498, −3.17916472240691115447459842518, −3.10361514593794498727934882345, −1.64499485444658532592961724782, −1.13630281923608134961414266624, 1.13630281923608134961414266624, 1.64499485444658532592961724782, 3.10361514593794498727934882345, 3.17916472240691115447459842518, 4.17540590010277769479068710498, 5.09767152865771135713722114004, 5.53926944552609927239784005951, 6.15149921353648408473937998277, 7.25043011485310189304278098122, 7.25526374973238832119327326179, 7.995271560601890934578164778827, 8.319202498085562194570074724910, 8.689362052218756979442645513614, 8.765082493453845884582129301720, 9.535443640693436142856681132436, 10.24020819636558820818875281638, 10.87642575347050855324411838616, 10.93880015048394908629159940051, 11.44715470203241819617330928940, 11.64448853237840978152738087403

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