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Properties

Label 4-8619-1.1-c1e2-0-0
Degree $4$
Conductor $8619$
Sign $1$
Analytic cond. $0.549554$
Root an. cond. $0.860999$
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  − 3·3-s − 3·4-s + 4·9-s + 9·12-s + 6·13-s + 5·16-s − 3·17-s + 14·23-s − 4·25-s + 2·29-s − 12·36-s − 18·39-s + 6·43-s − 15·48-s − 4·49-s + 9·51-s − 18·52-s + 2·53-s − 2·61-s − 3·64-s + 9·68-s − 42·69-s + 12·75-s − 4·79-s − 11·81-s − 6·87-s − 42·92-s + ⋯
L(s)  = 1  − 1.73·3-s − 3/2·4-s + 4/3·9-s + 2.59·12-s + 1.66·13-s + 5/4·16-s − 0.727·17-s + 2.91·23-s − 4/5·25-s + 0.371·29-s − 2·36-s − 2.88·39-s + 0.914·43-s − 2.16·48-s − 4/7·49-s + 1.26·51-s − 2.49·52-s + 0.274·53-s − 0.256·61-s − 3/8·64-s + 1.09·68-s − 5.05·69-s + 1.38·75-s − 0.450·79-s − 1.22·81-s − 0.643·87-s − 4.37·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8619\)    =    \(3 \cdot 13^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(0.549554\)
Root analytic conductor: \(0.860999\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8619,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3897473185\)
\(L(\frac12)\) \(\approx\) \(0.3897473185\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.23.ao_dq
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.ac_bi
31$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.31.a_bc
37$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.37.a_abe
41$C_2^2$ \( 1 - 60 T^{2} + p^{2} T^{4} \) 2.41.a_aci
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ag_di
47$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.47.a_aco
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ac_de
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.59.a_k
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.c_c
67$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.67.a_cg
71$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.71.a_acw
73$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.73.a_acc
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.e_ck
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.83.a_ba
89$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.89.a_k
97$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \) 2.97.a_aew
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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

−11.43937370705598108090338616877, −11.14806254035795264743981584195, −10.77429403207783901614870413658, −10.09605722941487764504994802742, −9.346779831482034811036937698120, −8.783489071344934016510690076816, −8.525001326249970503905195457168, −7.44352857996991884320614796215, −6.66725768558976199091482098115, −6.07760959298160390996407790776, −5.46156494845600818230038537459, −4.84598280143234120452733812979, −4.32073859087099149835822778458, −3.32207205156903976955607375908, −0.931125319871852265590842523801, 0.931125319871852265590842523801, 3.32207205156903976955607375908, 4.32073859087099149835822778458, 4.84598280143234120452733812979, 5.46156494845600818230038537459, 6.07760959298160390996407790776, 6.66725768558976199091482098115, 7.44352857996991884320614796215, 8.525001326249970503905195457168, 8.783489071344934016510690076816, 9.346779831482034811036937698120, 10.09605722941487764504994802742, 10.77429403207783901614870413658, 11.14806254035795264743981584195, 11.43937370705598108090338616877

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