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Properties

Label 4-68e2-1.1-c1e2-0-8
Degree $4$
Conductor $4624$
Sign $-1$
Analytic cond. $0.294830$
Root an. cond. $0.736873$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 4·5-s + 3·8-s − 6·9-s + 4·10-s − 4·13-s − 16-s + 2·17-s + 6·18-s + 4·20-s + 2·25-s + 4·26-s + 12·29-s − 5·32-s − 2·34-s + 6·36-s − 4·37-s − 12·40-s − 12·41-s + 24·45-s + 2·49-s − 2·50-s + 4·52-s + 12·53-s − 12·58-s − 20·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s − 2·9-s + 1.26·10-s − 1.10·13-s − 1/4·16-s + 0.485·17-s + 1.41·18-s + 0.894·20-s + 2/5·25-s + 0.784·26-s + 2.22·29-s − 0.883·32-s − 0.342·34-s + 36-s − 0.657·37-s − 1.89·40-s − 1.87·41-s + 3.57·45-s + 2/7·49-s − 0.282·50-s + 0.554·52-s + 1.64·53-s − 1.57·58-s − 2.56·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4624\)    =    \(2^{4} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(0.294830\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 4624,\ (\ :1/2, 1/2),\ -1)\)

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

−11.93623114442209313196485845061, −11.77779497781898747190652445510, −10.71734099889110000433050476633, −10.40227615321498563936322188024, −9.578693167598756855048719447448, −8.695686711870280818326611584019, −8.535338766321817121127070704904, −7.81910395523808377988054564239, −7.50901244752622102880251473780, −6.48235693417520450637241002127, −5.37611913672208711896121353581, −4.74199315541377008016560012773, −3.78911546643884845747068338400, −2.88331668043848910381979864823, 0, 2.88331668043848910381979864823, 3.78911546643884845747068338400, 4.74199315541377008016560012773, 5.37611913672208711896121353581, 6.48235693417520450637241002127, 7.50901244752622102880251473780, 7.81910395523808377988054564239, 8.535338766321817121127070704904, 8.695686711870280818326611584019, 9.578693167598756855048719447448, 10.40227615321498563936322188024, 10.71734099889110000433050476633, 11.77779497781898747190652445510, 11.93623114442209313196485845061

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