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Properties

Label 4-7803-1.1-c1e2-0-2
Degree $4$
Conductor $7803$
Sign $-1$
Analytic cond. $0.497526$
Root an. cond. $0.839854$
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  + 3-s − 4·4-s − 8·7-s + 9-s − 4·12-s − 2·13-s + 12·16-s − 2·19-s − 8·21-s − 25-s + 27-s + 32·28-s + 4·31-s − 4·36-s − 8·37-s − 2·39-s − 14·43-s + 12·48-s + 34·49-s + 8·52-s − 2·57-s + 16·61-s − 8·63-s − 32·64-s − 8·67-s + 4·73-s − 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 2·4-s − 3.02·7-s + 1/3·9-s − 1.15·12-s − 0.554·13-s + 3·16-s − 0.458·19-s − 1.74·21-s − 1/5·25-s + 0.192·27-s + 6.04·28-s + 0.718·31-s − 2/3·36-s − 1.31·37-s − 0.320·39-s − 2.13·43-s + 1.73·48-s + 34/7·49-s + 1.10·52-s − 0.264·57-s + 2.04·61-s − 1.00·63-s − 4·64-s − 0.977·67-s + 0.468·73-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7803\)    =    \(3^{3} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(0.497526\)
Root analytic conductor: \(0.839854\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 7803,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_1$ \( 1 - T \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−11.81013235135078842473393257576, −10.23827402727964604848387184191, −10.23122640158841960226025987177, −9.642727340101070443548067847258, −9.392750473138958647961697264371, −8.618966621189813626434174913816, −8.382258919016960888464931958518, −7.25132554126425354900557687617, −6.66463575826783700250676263362, −5.94155321226588727412442600100, −5.14306131647545403565477997005, −4.19114262883338708198809215469, −3.52343415212056409865262439727, −3.00632648437536992753290178800, 0, 3.00632648437536992753290178800, 3.52343415212056409865262439727, 4.19114262883338708198809215469, 5.14306131647545403565477997005, 5.94155321226588727412442600100, 6.66463575826783700250676263362, 7.25132554126425354900557687617, 8.382258919016960888464931958518, 8.618966621189813626434174913816, 9.392750473138958647961697264371, 9.642727340101070443548067847258, 10.23122640158841960226025987177, 10.23827402727964604848387184191, 11.81013235135078842473393257576

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