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Properties

Label 4-473200-1.1-c1e2-0-13
Degree $4$
Conductor $473200$
Sign $-1$
Analytic cond. $30.1716$
Root an. cond. $2.34368$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·5-s + 3·7-s + 3·8-s − 2·9-s + 2·10-s + 2·13-s − 3·14-s − 16-s + 2·18-s + 2·20-s − 25-s − 2·26-s − 3·28-s − 5·32-s − 6·35-s + 2·36-s + 4·37-s − 6·40-s + 4·45-s − 4·47-s + 6·49-s + 50-s − 2·52-s + 9·56-s − 4·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.13·7-s + 1.06·8-s − 2/3·9-s + 0.632·10-s + 0.554·13-s − 0.801·14-s − 1/4·16-s + 0.471·18-s + 0.447·20-s − 1/5·25-s − 0.392·26-s − 0.566·28-s − 0.883·32-s − 1.01·35-s + 1/3·36-s + 0.657·37-s − 0.948·40-s + 0.596·45-s − 0.583·47-s + 6/7·49-s + 0.141·50-s − 0.277·52-s + 1.20·56-s − 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(473200\)    =    \(2^{4} \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(30.1716\)
Root analytic conductor: \(2.34368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 473200,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.31.a_w
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.e_ac
53$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.53.a_o
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.e_eg
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.e_acg
71$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.71.a_aeo
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.e_g
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ae_ew
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.89.a_ck
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ai_gs
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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

−8.269682690619246794644739352656, −8.062032300044609257946916556183, −7.53642567009753475780215963243, −7.29145263875643546430560397006, −6.58821625207962417528600681663, −5.87422982899334878066388182128, −5.56370234044426669119725287739, −4.83136556543811478473932059788, −4.52102662865224081289750954538, −4.00242014811886142054616945899, −3.46457546231718726387570255821, −2.70524294515022445591456284384, −1.80141018999216604862217860039, −1.10433467477232311245779633925, 0, 1.10433467477232311245779633925, 1.80141018999216604862217860039, 2.70524294515022445591456284384, 3.46457546231718726387570255821, 4.00242014811886142054616945899, 4.52102662865224081289750954538, 4.83136556543811478473932059788, 5.56370234044426669119725287739, 5.87422982899334878066388182128, 6.58821625207962417528600681663, 7.29145263875643546430560397006, 7.53642567009753475780215963243, 8.062032300044609257946916556183, 8.269682690619246794644739352656

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