Properties

Label 4-9248-1.1-c1e2-0-2
Degree $4$
Conductor $9248$
Sign $1$
Analytic cond. $0.589660$
Root an. cond. $0.876295$
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 + 8-s + 2·9-s − 4·13-s + 16-s − 6·17-s + 2·18-s + 2·25-s − 4·26-s + 32-s − 6·34-s + 2·36-s − 14·49-s + 2·50-s − 4·52-s + 12·53-s + 64-s − 6·68-s + 2·72-s − 5·81-s − 12·89-s − 14·98-s + 2·100-s + 12·101-s − 4·104-s + 12·106-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 − 1.45·17-s + 0.471·18-s + 2/5·25-s − 0.784·26-s + 0.176·32-s − 1.02·34-s + 1/3·36-s − 2·49-s + 0.282·50-s − 0.554·52-s + 1.64·53-s + 1/8·64-s − 0.727·68-s + 0.235·72-s − 5/9·81-s − 1.27·89-s − 1.41·98-s + 1/5·100-s + 1.19·101-s − 0.392·104-s + 1.16·106-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: $\chi_{9248} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9248,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.413661249\)
\(L(\frac12)\) \(\approx\) \(1.413661249\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−11.71702842473540935817436808628, −11.09918432620632921533246736241, −10.59865118222208250930256448583, −9.941297116983973710463063002585, −9.479242170296714404313509737161, −8.720044960243225161536367200254, −8.074932466510213541223632938729, −7.22326385470593684044163515901, −6.91991985240048636916509773456, −6.21914096368307236962383159427, −5.31548945363166533632182177452, −4.65391290543775869547975796905, −4.11957344519078992021430964332, −2.98446406426293567318521383895, −2.00660833237422629012812096846, 2.00660833237422629012812096846, 2.98446406426293567318521383895, 4.11957344519078992021430964332, 4.65391290543775869547975796905, 5.31548945363166533632182177452, 6.21914096368307236962383159427, 6.91991985240048636916509773456, 7.22326385470593684044163515901, 8.074932466510213541223632938729, 8.720044960243225161536367200254, 9.479242170296714404313509737161, 9.941297116983973710463063002585, 10.59865118222208250930256448583, 11.09918432620632921533246736241, 11.71702842473540935817436808628

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