Properties

Label 4-135200-1.1-c1e2-0-5
Degree $4$
Conductor $135200$
Sign $1$
Analytic cond. $8.62046$
Root an. cond. $1.71349$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·5-s − 8-s + 4·9-s − 2·10-s + 4·13-s + 16-s − 4·17-s − 4·18-s + 2·20-s − 25-s − 4·26-s + 2·29-s − 32-s + 4·34-s + 4·36-s − 2·40-s + 8·41-s + 8·45-s − 10·49-s + 50-s + 4·52-s + 10·53-s − 2·58-s + 10·61-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s + 4/3·9-s − 0.632·10-s + 1.10·13-s + 1/4·16-s − 0.970·17-s − 0.942·18-s + 0.447·20-s − 1/5·25-s − 0.784·26-s + 0.371·29-s − 0.176·32-s + 0.685·34-s + 2/3·36-s − 0.316·40-s + 1.24·41-s + 1.19·45-s − 1.42·49-s + 0.141·50-s + 0.554·52-s + 1.37·53-s − 0.262·58-s + 1.28·61-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(135200\)    =    \(2^{5} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(8.62046\)
Root analytic conductor: \(1.71349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 135200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.594723038\)
\(L(\frac12)\) \(\approx\) \(1.594723038\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.ac_bi
31$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.31.a_au
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ai_ck
43$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.43.a_aq
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.53.ak_fa
59$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.59.a_acs
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.ak_es
67$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.67.a_by
71$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.71.a_bo
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.ak_by
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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

−9.347179855405233544904376539983, −8.891555216454554483498946537173, −8.594242606504294960977476680163, −7.87415637367127003226602367222, −7.45385864439445052849912105276, −6.87812345374716168263380118690, −6.41231748114411774436252215619, −6.05810036266471492576825148353, −5.42273665693311512142392085410, −4.66477500045914155874887391189, −4.11603275514460593439529740968, −3.46423065738529238809398406751, −2.46381795879742314340567797792, −1.85505053673419309584829247677, −1.07137365505227166891519930175, 1.07137365505227166891519930175, 1.85505053673419309584829247677, 2.46381795879742314340567797792, 3.46423065738529238809398406751, 4.11603275514460593439529740968, 4.66477500045914155874887391189, 5.42273665693311512142392085410, 6.05810036266471492576825148353, 6.41231748114411774436252215619, 6.87812345374716168263380118690, 7.45385864439445052849912105276, 7.87415637367127003226602367222, 8.594242606504294960977476680163, 8.891555216454554483498946537173, 9.347179855405233544904376539983

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