Properties

Label 4-28e2-1.1-c1e2-0-2
Degree $4$
Conductor $784$
Sign $1$
Analytic cond. $0.0499885$
Root an. cond. $0.472843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4·5-s + 2·7-s − 8-s − 2·9-s + 4·10-s − 4·13-s − 2·14-s + 16-s + 4·17-s + 2·18-s − 4·20-s + 8·23-s + 6·25-s + 4·26-s + 2·28-s − 4·29-s − 32-s − 4·34-s − 8·35-s − 2·36-s − 4·37-s + 4·40-s + 4·41-s + 16·43-s + 8·45-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.755·7-s − 0.353·8-s − 2/3·9-s + 1.26·10-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.471·18-s − 0.894·20-s + 1.66·23-s + 6/5·25-s + 0.784·26-s + 0.377·28-s − 0.742·29-s − 0.176·32-s − 0.685·34-s − 1.35·35-s − 1/3·36-s − 0.657·37-s + 0.632·40-s + 0.624·41-s + 2.43·43-s + 1.19·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.0499885\)
Root analytic conductor: \(0.472843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{784} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3130583414\)
\(L(\frac12)\) \(\approx\) \(0.3130583414\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( 1 + T \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−19.5800270198, −19.2726812580, −19.0818322220, −18.2120541317, −17.3843249773, −17.2128532162, −16.3370810014, −15.9707837644, −15.1678011465, −14.6077730657, −14.4714164826, −13.1074731398, −12.3052609901, −11.8508336106, −11.2313614144, −10.8509389842, −9.76554711946, −8.92832969723, −8.18805721451, −7.57571100089, −7.14438434722, −5.57928681743, −4.50185552371, −3.14137792992, 3.14137792992, 4.50185552371, 5.57928681743, 7.14438434722, 7.57571100089, 8.18805721451, 8.92832969723, 9.76554711946, 10.8509389842, 11.2313614144, 11.8508336106, 12.3052609901, 13.1074731398, 14.4714164826, 14.6077730657, 15.1678011465, 15.9707837644, 16.3370810014, 17.2128532162, 17.3843249773, 18.2120541317, 19.0818322220, 19.2726812580, 19.5800270198

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