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Properties

Label 4-9248-1.1-c1e2-0-1
Degree $4$
Conductor $9248$
Sign $1$
Analytic cond. $0.589660$
Root an. cond. $0.876295$
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 + 8-s − 2·9-s + 4·13-s + 16-s − 2·17-s − 2·18-s − 10·25-s + 4·26-s + 32-s − 2·34-s − 2·36-s − 8·37-s + 12·41-s + 2·49-s − 10·50-s + 4·52-s − 12·53-s − 8·61-s + 64-s − 2·68-s − 2·72-s + 4·73-s − 8·74-s − 5·81-s + 12·82-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2/3·9-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.471·18-s − 2·25-s + 0.784·26-s + 0.176·32-s − 0.342·34-s − 1/3·36-s − 1.31·37-s + 1.87·41-s + 2/7·49-s − 1.41·50-s + 0.554·52-s − 1.64·53-s − 1.02·61-s + 1/8·64-s − 0.242·68-s − 0.235·72-s + 0.468·73-s − 0.929·74-s − 5/9·81-s + 1.32·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9248\)    =    \(2^{5} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.589660\)
Root analytic conductor: \(0.876295\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9248,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.396783906\)
\(L(\frac12)\) \(\approx\) \(1.396783906\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−11.78868723809074966331144526462, −10.96199091064046403542442582980, −10.84894668716717132123408330664, −9.983024040967094825270124003185, −9.286071988264369821451329617999, −8.763699502340737285319959905301, −8.020764485917199688918114818768, −7.56004329429510605274038490355, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −5.66320470137327715389850473544, −4.65035915337384466130876980903, −3.90229547122613769308947697962, −3.14432806107716120656228384149, −1.95520344137895444953340586126, 1.95520344137895444953340586126, 3.14432806107716120656228384149, 3.90229547122613769308947697962, 4.65035915337384466130876980903, 5.66320470137327715389850473544, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 7.56004329429510605274038490355, 8.020764485917199688918114818768, 8.763699502340737285319959905301, 9.286071988264369821451329617999, 9.983024040967094825270124003185, 10.84894668716717132123408330664, 10.96199091064046403542442582980, 11.78868723809074966331144526462

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