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Properties

Label 4-114e2-1.1-c1e2-0-3
Degree $4$
Conductor $12996$
Sign $1$
Analytic cond. $0.828636$
Root an. cond. $0.954093$
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-s + 3-s + 4·5-s − 6-s − 6·7-s + 8-s − 4·10-s + 4·11-s + 7·13-s + 6·14-s + 4·15-s − 16-s − 8·19-s − 6·21-s − 4·22-s + 4·23-s + 24-s + 5·25-s − 7·26-s − 27-s − 4·29-s − 4·30-s + 2·31-s + 4·33-s − 24·35-s + 14·37-s + 8·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1.78·5-s − 0.408·6-s − 2.26·7-s + 0.353·8-s − 1.26·10-s + 1.20·11-s + 1.94·13-s + 1.60·14-s + 1.03·15-s − 1/4·16-s − 1.83·19-s − 1.30·21-s − 0.852·22-s + 0.834·23-s + 0.204·24-s + 25-s − 1.37·26-s − 0.192·27-s − 0.742·29-s − 0.730·30-s + 0.359·31-s + 0.696·33-s − 4.05·35-s + 2.30·37-s + 1.29·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12996\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.828636\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9792291713\)
\(L(\frac12)\) \(\approx\) \(0.9792291713\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_l
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ah_bk
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_an
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.31.ac_cl
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.37.ao_et
41$C_2^2$ \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_az
43$C_2^2$ \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.43.h_g
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_abr
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_abl
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ax
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.ab_aci
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.67.d_acg
71$C_2^2$ \( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_acp
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.73.ad_acm
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.79.f_acc
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_jb
97$C_2^2$ \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.97.k_d
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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

−13.52075601597673538336852622496, −13.37268083446081823670774685786, −12.93675058068021834249048812864, −12.81662285296933955415682074037, −11.64041903378439221298354564433, −11.07956292177406139206527796372, −10.44568889777340241685417550840, −9.759312975536325898130913456397, −9.698975745534639055029712035201, −9.197588431634538098208427051470, −8.609816077197101248862353423767, −8.356139788180140540496179767920, −6.93071666202233534660996397495, −6.62925058745666810014651140038, −5.95861093515340541713444358495, −5.91093817892060880743479689853, −4.25104104797604112613275752221, −3.56266624530711247778126453860, −2.70076862898533993905049973728, −1.54913072358068928721326785821, 1.54913072358068928721326785821, 2.70076862898533993905049973728, 3.56266624530711247778126453860, 4.25104104797604112613275752221, 5.91093817892060880743479689853, 5.95861093515340541713444358495, 6.62925058745666810014651140038, 6.93071666202233534660996397495, 8.356139788180140540496179767920, 8.609816077197101248862353423767, 9.197588431634538098208427051470, 9.698975745534639055029712035201, 9.759312975536325898130913456397, 10.44568889777340241685417550840, 11.07956292177406139206527796372, 11.64041903378439221298354564433, 12.81662285296933955415682074037, 12.93675058068021834249048812864, 13.37268083446081823670774685786, 13.52075601597673538336852622496

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