Properties

Label 4-91e2-1.1-c1e2-0-3
Degree $4$
Conductor $8281$
Sign $1$
Analytic cond. $0.528003$
Root an. cond. $0.852431$
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  + 2·2-s + 3·3-s − 4-s − 3·5-s + 6·6-s + 4·7-s − 8·8-s + 3·9-s − 6·10-s + 3·11-s − 3·12-s − 2·13-s + 8·14-s − 9·15-s − 7·16-s − 4·17-s + 6·18-s + 19-s + 3·20-s + 12·21-s + 6·22-s − 24·24-s + 5·25-s − 4·26-s − 4·28-s − 7·29-s − 18·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s − 1/2·4-s − 1.34·5-s + 2.44·6-s + 1.51·7-s − 2.82·8-s + 9-s − 1.89·10-s + 0.904·11-s − 0.866·12-s − 0.554·13-s + 2.13·14-s − 2.32·15-s − 7/4·16-s − 0.970·17-s + 1.41·18-s + 0.229·19-s + 0.670·20-s + 2.61·21-s + 1.27·22-s − 4.89·24-s + 25-s − 0.784·26-s − 0.755·28-s − 1.29·29-s − 3.28·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8281\)    =    \(7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.528003\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{91} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8281,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.924062198\)
\(L(\frac12)\) \(\approx\) \(1.924062198\)
\(L(\frac{3}{2})\) 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)$
bad7$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−14.46585895822143833311136832592, −14.11229235269877279002746093897, −13.31546639609939714767137797220, −13.16586848332719005533561895932, −12.31949065152968956766145001265, −11.95064385211965783578853288240, −11.42096805350290347665023111908, −10.90733260216413571841262765518, −9.621496200049852987512287090668, −9.145286917569303267357955186980, −8.809710892776191299255148903689, −8.239790492814534830455196396157, −7.87691833304512702463473280831, −7.13166786646256717953807301980, −5.99997245797369236524473297651, −5.01391016386276166050628934385, −4.58136042881269318009410929438, −3.86124221618274047271406154688, −3.55245046201534053927398399933, −2.46899221090451149875695901858, 2.46899221090451149875695901858, 3.55245046201534053927398399933, 3.86124221618274047271406154688, 4.58136042881269318009410929438, 5.01391016386276166050628934385, 5.99997245797369236524473297651, 7.13166786646256717953807301980, 7.87691833304512702463473280831, 8.239790492814534830455196396157, 8.809710892776191299255148903689, 9.145286917569303267357955186980, 9.621496200049852987512287090668, 10.90733260216413571841262765518, 11.42096805350290347665023111908, 11.95064385211965783578853288240, 12.31949065152968956766145001265, 13.16586848332719005533561895932, 13.31546639609939714767137797220, 14.11229235269877279002746093897, 14.46585895822143833311136832592

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