Properties

Label 4-91e2-1.1-c1e2-0-2
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s + 3·5-s − 4·7-s + 3·9-s − 9·11-s + 12-s − 2·13-s + 3·15-s − 3·16-s + 12·17-s − 3·19-s + 3·20-s − 4·21-s + 25-s + 8·27-s − 4·28-s − 3·29-s − 3·31-s − 9·33-s − 12·35-s + 3·36-s − 2·39-s − 9·41-s − 11·43-s − 9·44-s + 9·45-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 1.34·5-s − 1.51·7-s + 9-s − 2.71·11-s + 0.288·12-s − 0.554·13-s + 0.774·15-s − 3/4·16-s + 2.91·17-s − 0.688·19-s + 0.670·20-s − 0.872·21-s + 1/5·25-s + 1.53·27-s − 0.755·28-s − 0.557·29-s − 0.538·31-s − 1.56·33-s − 2.02·35-s + 1/2·36-s − 0.320·39-s − 1.40·41-s − 1.67·43-s − 1.35·44-s + 1.34·45-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

\(L(1)\) \(\approx\) \(1.236587578\)
\(L(\frac12)\) \(\approx\) \(1.236587578\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 15 T + 122 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.20160350118561470449306431338, −13.55440261211125451535870249322, −13.48443661857895039273340476120, −12.70550912050822825842063304307, −12.59200406575555224235853639917, −11.98637722059681998716422776831, −10.73801558792837463651405583694, −10.32864071308234210480674182498, −9.970315481442408795627502753107, −9.803648140997261775495606943072, −8.996556265439986671266390640436, −8.073405316114595917085535553671, −7.58777677589498127478623729835, −6.98248780947366634934788759535, −6.30089923852536836045224263643, −5.32886254199497551754572328547, −5.29003435010259500926597715634, −3.61133977084879571321264428147, −2.80303467746832941482307451990, −2.18642467623826182589463232876, 2.18642467623826182589463232876, 2.80303467746832941482307451990, 3.61133977084879571321264428147, 5.29003435010259500926597715634, 5.32886254199497551754572328547, 6.30089923852536836045224263643, 6.98248780947366634934788759535, 7.58777677589498127478623729835, 8.073405316114595917085535553671, 8.996556265439986671266390640436, 9.803648140997261775495606943072, 9.970315481442408795627502753107, 10.32864071308234210480674182498, 10.73801558792837463651405583694, 11.98637722059681998716422776831, 12.59200406575555224235853639917, 12.70550912050822825842063304307, 13.48443661857895039273340476120, 13.55440261211125451535870249322, 14.20160350118561470449306431338

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